CHAPTER 9 Topics in Analytic Geometry - crunchy...

C H A P T E R 9Topics in Analytic Geometry

Section 9.1 Circles and Parabolas . . . . . . . . . . . . . . . . . . . . 772

Section 9.2 Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . 784

Section 9.3 Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . 795

Section 9.4 Rotation and Systems of Quadratic Equations . . . . . . . 807

Section 9.5 Parametric Equations . . . . . . . . . . . . . . . . . . . . 825

Section 9.6 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . 833

Section 9.7 Graphs of Polar Equations . . . . . . . . . . . . . . . . . 845

Section 9.8 Polar Equations of Conics . . . . . . . . . . . . . . . . . 854

Review Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863

Practice Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886

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C H A P T E R 9Topics in Analytic Geometry

Section 9.1 Circles and Parabolas

772

! A parabola is the set of all points that are equidistant from a fixed line (directrix) and a fixed point(focus) not on the line.

! The standard equation of a parabola with vertex and

(a) Vertical axis and directrix is

(b) Horizontal axis and directrix is

! The tangent line to a parabola at a point makes equal angles with

(a) the line through and the focus.

(b) the axis of the parabola.

P

P

(y ! k)2 " 4p(x ! h), p # 0.

x " h ! p y " k

(x ! h)2 " 4p( y ! k), p # 0.

y " k ! p x " h

!h, k"

!x, y"

Vocabulary Check1. conic section 2. locus 3. circle, center

4. parabola, directrix, focus 5. vertex 6. axis

7. tangent

1.

x2 $ y2 " 18

x2 $ y2 " !#18"2 2.

x2 $ y2 " 32

x2 $ y2 " !4#2 "2

3.

!x ! 3"2 $ ! y ! 7"2 " 53

!x ! h"2 $ ! y ! k"2 " r2

" #4 $ 49 " #53

Radius " #!3 ! 1"2 $ !7 ! 0"2 4.

!x ! 6"2 $ ! y $ 3"2 " 113

!x ! h"2 $ ! y ! k"2 " r2

" #64 $ 49 " #113

Radius " #$6 ! !!2"%2 $ $!3 ! 4%2

5.

!x $ 3"2 $ ! y $ 1"2 " 7

!x ! h"2 $ ! y ! k"2 " r2

Diameter " 2#7 ! radius " #7 6.

!x ! 5"2 $ ! y $ 6"2 " 12

!x ! h"2 $ ! y ! k"2 " r2

Diameter " 4#3 ! radius " 2#3

7.

Center:

Radius: 7

!0, 0"

x2 $ y2 " 49 8.

Center:

Radius: 1

!0, 0"

x2 $ y2 " 1 9.

Center:

Radius: 4

!!2, 7"

!x $ 2"2 $ !y ! 7"2 " 16

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Section 9.1 Circles and Parabolas 773

10.

Center:

Radius: 6

!!9, !1"

!x $ 9"2 $ !y $ 1"2 " 36 11.

Center:

Radius: #15

!1, 0"

!x ! 1"2 $ y2 " 15 12.

Center:

Radius: #24 " 2#6

!0, !12"

x2 $ !y $ 12"2 " 24

13.

Center:

Radius: 2

!0, 0"

x2 $ y2 " 4

14

x2 $14

y2 " 1 14.

Center:

Radius: 3

!0, 0"

x2 $ y2 " 9

19

x2 $19

y2 " 1 15.

Center:

Radius:#32

!0, 0"

x2 $ y2 "34

43

x2 $43

y2 " 1

16.

Center:

Radius:#23

!0, 0"

x2 $ y2 "29

92

x2 $92

y2 " 1 17.

Center:

Radius: 1

!1, !3"

!x ! 1"2 $ !y $ 3"2 " 1

!x2 ! 2x $ 1" $ !y2 $ 6y $ 9" " !9 $ 1 $ 9

18.

Center:

Radius: 3

!5, 3"

!x ! 5"2 $ !y ! 3"2 " 9

!x2 ! 10x $ 25" $ !y2 ! 6y $ 9" " !25 $ 25 $ 9 19.

Center:

Radius: 1

!!32, 3"

!x $ 32"2

$ ! y ! 3"2 " 1

4!x $ 32"2

$ 4!y ! 3"2 " 4

4!x2 $ 3x $ 94" $ 4!y2 ! 6y $ 9" " !41 $ 9 $ 36

20.

Center:

Radius: 103

!!3, 2"

!x $ 3"2 $ !y ! 2"2 " 1009

9!x $ 3"2 $ 9!y ! 2"2 " 100

9!x2 $ 6x $ 9" $ 9!y2 ! 4y $ 4" " !17 $ 81 $ 36

21.

Center:

Radius: 4

!0, 0"

x2 $ y2 " 16

!1!2!3!5 1 2 3 5

!2!3

!5

123

5

x

y x2 " 16 ! y2 22.

Center:

Radius: 9

!0, 0"

x2 $ y2 " 81

!2!4!6!10 2 4 6 8 10

!4!6!8

!10

2468

10

x

y y2 " 81 ! x2

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774 Chapter 9 Topics in Analytic Geometry

23.

Center:

Radius: 3

!1!2!3!5!6!7 2 3

!2!3!4

!6!7

23

x

y!!2, !2"

!x $ 2"2 $ ! y $ 2"2 " 9

!x2 $ 4x $ 4" $ ! y2 $ 4y $ 4" " 1 $ 4 $ 4

x2 $ 4x $ y2 $ 4y ! 1 " 0 24.

Center:

Radius: 2!1!2 2 51 4 6 7 8

!2!3!4!5!6!7!8!9

1x

y!3, !3"

!x ! 3"2 $ ! y $ 3"2 " 4

!x2 ! 6x $ 9" $ ! y2 $ 6y $ 9" " !14 $ 9 $ 9

x2 ! 6x $ y2 $ 6y $ 14 " 0

25.

Center:

Radius: 5

!7, !4"

!x ! 7"2 $ ! y $ 4"2 " 25

!x2 ! 14x $ 4" $ ! y2 $ 8y $ 16" " !40 $ 49 $ 16

!2 4 6 8 10

!4!6!8

!10!12!14

246

x

y

14 16 18

x2 ! 14x $ y2 $ 8y $ 40 " 0

26.

Center:

Radius: 2

!1!2!3!4!5!6!7!8 1 2

123456789

x

y!!3, 6"

!x $ 3"2 $ ! y ! 6"2 " 4

!x2 $ 6x $ 9" $ ! y2 ! 12y $ 36" " !41 $ 9 $ 36

x2 $ 6x $ y2 ! 12y $ 41 " 0 27.

Center:

Radius: 6

!2!4!8!10 2 4 6 8 10

!4

!8!10

24

810

x

y!!1, 0"

!x $ 1"2 $ y2 " 36

!x2 $ 2x $ 1" $ y2 " 35 $ 1

x2 $ 2x $ y2 ! 35 " 0

28.

Center:

Radius: 4!1!2!3!4!5 1 2 3 4 5

!2!3!4!5!6!7!8

1x

y!0, !5"

x2 $ ! y2 $ 5"2 " 16

x2 $ ! y2 $ 10y $ 25" " !9 $ 25

x2 $ y2 $ 10y $ 9 " 0 29. intercepts:

intercepts:

!2, 0"

x " 2

!x ! 2"2 " 0

!x ! 2"2 $ !0 $ 3"2 " 9x-

!0, !3 ± #5 " y " !3 ± #5

! y $ 3"2 " 5

4 $ ! y $ 3"2 " 9

!0 ! 2"2 $ ! y $ 3"2 " 9y-

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30. intercepts:

intercepts:

!!2, 0", !!8, 0"

x " !8, !2

x $ 5 " ±3

!x $ 5"2 " 9

!x $ 5"2 $ 16 " 25

!x $ 5"2 $ !0 ! 4"2 " 25x-

!0, 4"

y " 4

! y ! 4"2 " 0

!0 $ 5"2 $ ! y ! 4"2 " 25y- 31. intercepts: Let

intercepts: Let

!1 ± 2#7, 0" x " 1 ± 2#7

x ! 1 " ±#28

!x ! 1"2 " 28

x2 ! 2x $ 1 " 27 $ 1

x2 ! 2x ! 27 " 0

y " 0.x-

!0, 9", !0, !3" y " 9, !3

y ! 3 " ±6

! y ! 3"2 " 36

y2 ! 6y $ 9 " 27 $ 9

y2 ! 6y ! 27 " 0

x " 0.y-

32. intercepts: Let

No solution

No intercepts

intercepts: Let

!!4 ± #7, 0" x " !4 ± #7

x $ 4 " ±#7

!x $ 4"2 " 7

x2 $ 8x $ 16 " !9 $ 16

x2 $ 8x $ 9 " 0

y " 0.x-

y-

y2 $ 2y $ 9 " 0

x " 0.y- 33. intercepts:

No solution

No intercepts

intercepts:

!6 ± #7, 0" x " 6 ± #7

x ! 6 " ±#7

!x ! 6"2 " 7

!x ! 6"2 $ !0 $ 3"2 " 16x-

y-

" !20

! y $ 3"2 " 16 ! 36

!0 ! 6"2 $ ! y $ 3"2 " 16y-

34. intercepts:

No solution

No intercepts

intercepts:

No solution

No interceptsx-

" !60

!x $ 7"2 " 4 ! 64

!x $ 7"2 $ !0 ! 8"2 " 4x-

y-

" !45

! y ! 8"2 " 4 ! 49

!0 $ 7"2 $ ! y ! 8"2 " 4y- 35. (a) Radius: 81; Center:

(b) The distance from to is

Yes, you would feel the earthquake.

(c)

You were miles from the outerboundary.

81 ! 75 " 6

x

y

!40 40

!40

40 (60, 45)

x2 + y2 = 812

#602 $ 452 " #5625 " 75 miles.

!0, 0"!60, 45"x2 $ y2 " 812 " 6561

!0, 0"

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36. (a)

r & 23.937 feet

r "#1800%

r2 "1800

%

Area " %r2 " 1800 (b)

longer radius27.640 ! 23.937 & 3.703

R "#2400%

& 27.640 feet

%R2 " 2400

37.

Vertex:

Opens to the left since isnegative.

Matches graph (e).

p

!0, 0"

y2 " !4x 38.

Vertex:

Opens upward

Matches graph (b).

p " 12 > 0

!0, 0"

x2 " 2y 39.

Vertex:

Opens downward since isnegative.

Matches graph (d).

p

!0, 0"

x2 " !8y

40.

Vertex:

Opens to the left

Matches graph (f).

p " !3 < 0

!0, 0"

y2 " !12x 41.

Vertex:

Opens to the right since ispositive.

Matches graph (a).

p

!3, 1"

(y ! 1)2 " 4(x ! 3) 42.

Vertex:

Opens downward

Matches graph (c).

p " ! 12 < 0

!!3, 1"

!x $ 3"2 " !2!y ! 1"

43. Vertex:

Graph opens upward.

Point on graph:

! x2 " 32 y.

Thus, x2 " 4!38"y ! y " 2

3 x2

38 " p

9 " 24p

32 " 4p!6"

!3, 6"

x2 " 4py

!0, 0" ! h " 0, k " 0 44. Point:

y2 " !18x

x " ! 118 y2

! 118 " a

!2 " a!6"2

x " ay2

!!2, 6" 45. Vertex:

Focus:

x2 " !6y

x2 " 4!!32"y

!x ! h"2 " 4p!y ! k"

!0, !32" ! p " !3

2

!0, 0" ! h " 0, k " 0

46. Focus:

y2 " 10x

y2 " 4px " 4! 52"x

! 52, 0" ! p " 5

2 47. Vertex:

Focus:

y2 " !8x

y2 " 4!!2"x

!y ! k"2 " 4p!x ! h"

!!2, 0" ! p " !2

!0, 0" ! h " 0, k " 0 48. Focus:

x2 " 4y

x2 " 4py " 4!1"y

!0, 1" ! p " 1

49. Vertex:

Directrix:

x2 " 4y or y " 14x2

!x ! 0"2 " 4!1"!y ! 0"

!x ! h"2 " 4p!y ! k"

y " !1 ! p " 1

!0, 0" ! h " 0, k " 0 50. Directrix:

x2 " !12y

x2 " 4py

y " 3 ! p " !3 51. Vertex:

Directrix:

y2 " !8x

y2 " 4px

x " 2 ! p " !2

!0, 0" ! h " 0, k " 0

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52. Directrix:

y2 " 12x

y2 " 4px

x " !3 ! p " 3

53. Vertex:

Horizontal axis and passes through the point

y2 " 9x

y2 " 4!94"x

36 " 16p ! p " 94

62 " 4p!4"

y2 " 4px

!y ! 0"2 " 4p!x ! 0"

!y ! k"2 " 4p!x ! h"

!4, 6"

!0, 0" ! h " 0, k " 0 54. Vertical axis

Passes through

x2 " !3y

x2 " 4!!34"y

9 " !12p ! p " !34

!!3"2 " 4p!!3"

x2 " 4py

!!3, !3"

55.

Vertex:

Focus:

Directrix:

–1

1

2

3

4

5

–3 –2 2 3

y

x

y " !12

!0, 12"!0, 0"

x2 " 2y " 4! 12 "y; p " 1

2

y " 12 x2 56.

Vertex:

Focus:

Directrix:

!1!2 1 2

!2

!3

!4

x

y

y " 116

!0, ! 116"

!0, 0"

x2 " !14 y " 4!! 1

16"y, p " ! 116

y " !4x2 57.

Vertex:

Focus:

Directrix:

–6 –5 –4 –3 –2 –1 1 2

–4

–3

3

4

y

x

x " 32

!!32, 0"

!0, 0"

y2 " 4!!32"x; p " !3

2

y2 " !6x

58.

Vertex:

Focus:

Directrix:

!2 2 4 6

4

y

x

x " !34

!34, 0"!0, 0"

y2 " 4!34"x; p " 3

4

y2 " 3x 59.

Vertex:

Focus:

Directrix:

x

y

!4!6!8 4 6 8!2

!4

!6

!8

!10

4

6

2

y " 2

!0, !2"

!0, 0"

x2 " 4!!2"y; p " !2

x2 $ 8y " 0 60.

Vertex:

Focus:

Directrix:

–5 –4 –3 –2 –1 1

–3

–2

2

3

x

y

x " 14

!!14, 0"

!0, 0"

y2 " 4!!14"x, p " !1

4

y2 " !x

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61.

Vertex:

Focus:

Directrix: y " !1

!!1, !5"x

y

!2

!4

!6

!8

!10

!12

4

2

2

!!1, !3"

h " !1, k " !3, p " !2

!x $ 1"2 " 4!!2"!y $ 3"

!x $ 1"2 $ 8!y $ 3" " 0 62.

Vertex:

Focus:

Directrix:

x

y

!1!2 1 2 3 4 5 6!1

!2

!3

!4

!5

!6

!7

1

x " 214

!5 ! 14, !4" " !19

4 , !4"!5, !4"

!y $ 4"2 " !!x ! 5" " 4!!14"!x ! 5"

!x ! 5" $ !y $ 4"2 " 0

63.

Vertex:

Focus:

Directrix: x " 0

!!4, !3"–10 –8 –6 –4

–8

–6

–4

–2

2

x

y!!2, !3"

!y $ 3"2 " 4!!2"!x $ 2"; p " !2

y2 $ 6y $ 8x $ 25 " 0 64.

Vertex:

Focus:

Directrix: x " !2

!0, 2"

–4 2 4

–2

4

6

x

y!!1, 2"

!y ! 2"2 " 4!x $ 1"; p " 1

y2 ! 4y ! 4x " 0

65.

Vertex:

Focus:

Directrix:

1 32!1!3 !2!4!5

!2

3

4

5

6

x

yy " 1

!!32, 2 $ 1" " !!3

2, 3"!!3

2, 2"h " !3

2, k " 2, p " 1!!x $ 32"2

" 4!y ! 2" 66.

Vertex:

Focus:

Directrix:

!1!2!3

!1

3

4

1 2x

yy " 0

!!12, 1 $ 1" " !!1

2, 2"!!1

2, 1"!x $ 1

2"2" 4!y ! 1" ! p " 1

67.

Vertex:

Focus:

Directrix: y " 0

!1, 2"

!1, 1"

h " 1, k " 1, p " 1

!x ! 1"2 " 4!1"!y ! 1"

–2 2 4

2

4

6

x

y 4y ! 4 " !x ! 1"2

y " 14!x2 ! 2x $ 5" 68.

Vertex:

Focus:

Directrix: x " 7

!9, !1"

!2

!4

!6

2

4

6

2 4 6 10 12x

y!8, !1"

!y $ 1"2 " 4!1"!x ! 8"

y2 $ 2y $ 1 " 4x ! 33 $ 1

4x ! y2 ! 2y ! 33 " 0

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69.

Vertex:

Focus:

Directrix:

2!1!3!4!6

!2

!3

3

2

1

4

5

x

yy " 52

!!2, 1 ! 32" " !!2, !1

2"!!2, 1"

!x $ 2"2 " 4!!32"!y ! 1"

!x $ 2"2 " !6!y ! 1"

x2 $ 4x $ 4 " !6y $ 2 $ 4 " !6y $ 6

x2 $ 4x $ 6y ! 2 " 0 70.

Vertex:

Focus:

Directrix: y " 1

!1,!3"!1!2

!2

2

!3!4!5!6!7!8

2 3 4 5 6!3!4x

y!1, !1"

!x ! 1"2 " !8!y $ 1" " 4!!2"!y $ 1"

x2 ! 2x $ 1 " !8y ! 9 $ 1

x2 ! 2x $ 8y $ 9 " 0

71.

Vertex:

Focus:

Directrix:

To use a graphing calculator, enter:

y2 " !12 ! #1

4 ! x

y1 " !12 $ #1

4 ! x

x " 12

!0, !12"

!1 1!2

!2

1

2

!3x

y!14, !1

2"h " 1

4, k " !12, p " !1

4

!y $ 12"2 " 4!!1

4"!x ! 14"

y2 $ y $ 14 " !x $ 1

4

y2 $ x $ y " 0 72.

Vertex:

Focus:

Directrix: x " !2

!0, 0"

!4

!4

!2

2

4

6

!6

2 4 6 8x

y!!1, 0"

y2 " 4x $ 4 " 4!1"!x $ 1"

y2 ! 4x ! 4 " 0

73. Vertex:opens downward

Passes through:

!x ! 3"2 " !!y ! 1"

" !!x ! 3"2 $ 1

" !x2 $ 6x ! 8

y " !!x ! 2"!x ! 4"

!2, 0", !4, 0"

!3, 1", 74. Vertex:

Passes through:

!y ! 3"2 " !2!x ! 5"

p " ! 12

1 " 4p!4.5 ! 5"

!y ! 3"2 " 4p!x ! 5"

!y ! k"2 " 4p!x ! h"

!4.5, 4"

k " 3!5, 3" ! h " 5, 75. Vertex:opens to the right

Focus:

y2 " 2!x $ 2"

y2 " 4!12"!x $ 2"

12 " p

!!32, 0"

!!2, 0",

76. Vertex:

Focus:

!x ! 3"2 " 3!y $ 3"

!x ! h"2 " 4p!y ! k"

!3, !94" ! p " 3

4

k " !3!3, !3" ! h " 3, 77. Vertex:

Focus:

Horizontal axis:

!y ! 2"2 " !8!x ! 5"

!y ! 2"2 " 4!!2"!x ! 5"

p " 3 ! 5 " !2

!3, 2"

!5, 2"

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81. Focus:

Directrix:

Horizontal axis

Vertex:

!y ! 2"2 " 8x

!y ! 2"2 " 4!2"!x ! 0"

p " 2 ! 0 " 2

!0, 2"

x " !2

!2, 2" 82. Focus:

Directrix:

Vertex:

x2 " !8!y ! 2"

x2 " 4!!2"!y ! 2"

!0, 2"

y " 4 ! p " !2

!0, 0"

83.

The point of tangency is!2, 4".

y2 " !#8x

!3

!6 6

5 y1 " #8x

y2 " 8x and x ! $ 2y3 " x $ 2

y2 ! 8x " 0x and 3x ! y $ 2 " 0 84.

The point of tangency is!6, !3".

!4

!4 8

4 y1 " ! 112 x2

12y " !x2 y2 " 3 ! x

x2 $ 12y " 0 and x $ y ! 3 " 0

85. focus:

Following Example 4, we find the intercept

Tangent line

Let intercept !2, 0".y " 0 ! x " 2 ! x-

y " 4x ! 8,

m "8 ! !!8"

4 ! 0" 4

b " !8!12

! b "172

!d1 " d2

d2 "#!4 ! 0"2 $ '8 !12(

2"

172

d1 "12

! b

!0, b".y-

'0, 12(!4, 8", p "

12

,x2 " 2y,

78. Vertex:

Focus:

!x $ 1"2 " !8!y ! 2"

!x $ 1"2 " 4!!2"!y ! 2"

!x ! h"2 " 4p!y ! k"

!!1, 0" ! p " !2

k " 2!!1, 2" ! h " !1, 79. Vertex:

Directrix:

Vertical axis

x2 " 8!y ! 4"

!x ! 0"2 " 4!2"!y ! 4"

p " 4 ! 2 " 2

y " 2

!0, 4" 80. Vertex:

Directrix:

!y ! 1"2 " !12!x $ 2"

!y ! 1"2 " 4!!3"!x ! !!2""

!y ! k"2 " 4p!x ! h"

x " 1 ! p " !3

k " 1!!2, 1" ! h " !2,

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86.

Focus:

Tangent line:

-intercept: '!32

, 0(x

y " !3x !92

! 6x $ 2y $ 9 " 0

m "!!9)2" ! !9)2"

0 $ 3" !3

b " !92

12

! b " 5

d2 "#!!3 ! 0"2 $ '92

!12(

2

" 5

d1 "12

! b

'0, 12(

p "12

4'12(y " x2

2y " x2 87.

Focus:

Following Example 4, we find the intercept

Let intercept '!12

, 0(.y " 0 ! x " !12

! x-

y " 4x $ 2

m "!2 ! 2!1 ! 0

" 4

b " 2!18

$ b "178

!d1 " d2

d2 "#!!1 ! 0"2 $ '!2 $18(

2"

178

d1 "18

$ b

!0, b".y-

'0, !18(

! p " !18

y " !2x2 ! x2 " !12

y " 4'!18(y

88.

Focus:

Intercept: !1, 0"

y " !8x $ 8

m "!8 ! 82 ! 0

" !8

d1 " d2 ! 18

$ b "658

! b " 8

d2 "#!2 ! 0"2 $ '!8 $18(

2"

658

d1 "18

$ b

'0, !18(

x2 " !12

y " 4'!18(y ! p " !

18

y " !2x2, !2, !8" 89.

is a maximum of $23,437.50 when televisions.

00 250

25,000

x " 125R

R " 375x !32

x2

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90. (a)

y "x2

12,288

x2 " 4!3072"y

3072 " p

1024 "13

p

322 " 4p' 112(

x2 " 4py 91. (a)

(b) When

Depth: inches83

y "166

"83

.

6y " 16

x " 4,

!or y2 " 6x"

x2 " 4'32(y " 6y

x

y

!1!2!3!4 1 2 3 4!1

!2

1

2

4

5

6

320,( (

8 in.

x2 " 4py, p "32

(b)

x & 22.6 feet

512 " x2

12,288

24" x2

124

"x2

12,288

92. on parabola

The wire should be insertedinches from the bottom.9

4

p " 3616 " 9

4

36 " 4p!4"

x

y

!2!4!6 2 4 6!2

2

6

8

10

(6, 4)

x2 " 4py, !6, 4" 93. (a)

(c)

y

x

(!640, 152) (640, 152)

(b)

y " 1951,200 x2

p " 12,80019

6402 " 4p!152"

x2 " 4py

x 0 200 400 500 600

y 0 14.84 59.38 92.77 133.59

94. (a) passes through point

or

(b) !0.1 " ! 1640 x2 ! x " 8 feet

y " ! 1640 x2 x2 " !640y

x2 " 4!!160"y

256 " 4p!!25" ! p " !160

!16, !25".x2 " 4py 95. Vertex:

Point:

y2 " 640x

y2 " 4!160"x

8002 " 4p!1000% ! p " 160

!1000, 800"

y2 " 4px

!0, 0%

96. (a)

(b)

! ! 0"x2 " !16,400!y ! 4100"

!x ! 0"2 " 4!!4100"!y ! 4100"

p " !4100, !h, k" " !0, 4100"

V " 17,500#2 mi)hr & 24,750 mi)hr 97.

(a)

(b) The highest point is at Thedistance is the -intercept of feet.&15.69x

!6.25, 7.125".

00 16

10

y " !0.08x2 $ x $ 4

!12.5y $ 89.0625 " x2 ! 12.5x $ 39.0625

!12.5! y ! 7.125" " !x ! 6.25"2

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98. (a)

(b)

! x & 69.3 ft

y " 0 " !164

x2 $ 75 ! x2 " 75!64"

" !164

x2 $ 75

" !16x2

322 $ 75

y " !16x2

v2 $ s

x2 " !116

v2!y ! s" 99. The slope of the line joining and the centeris The slope of the tangent line at is Thus,

3x ! 4y " 25, tangent line.

4y $ 16 " 3x ! 9

y $ 4 "34

!x ! 3"

34.!3, !4"!4

3.!3, !4"

100. The slope of the line joining and the center is The slope of the tangent line at is Thus,

5x ! 12y $ 169 " 0, tangent line.

12y ! 144 " 5x $ 25

y ! 12 "512

!x $ 5"

512.!!5, 12"

!125 .

!!5, 12" 101. The slope of the line joining and the center is The slope ofthe tangent line is Thus,

#2x ! 2y " 6#2, tangent line.

2y $ 4#2 " #2x ! 2#2

y $ 2#2 "#22

!x ! 2"

1)#2 " #2)2.!!2#2 ")2 " !#2.

!2, !2#2 "

102. The slope of the line joining and the center is The slope of the tangent line is Thus,

#5x ! y $ 12 " 0, tangent line.

y ! 2 " #5x $ 10

y ! 2 " #5!x $ 2#5 "#5.

2)!!2#5 " " !1)#5.!!2#5, 2"

103. False. The center is !0, !5". 104. True 105. False. A circle is a conic section.

106. False. A parabola cannotintersect its directrix or focus.

107. True 108. False. The directrix is below the axis.x-

y " !14

109. Answers will vary. See the reflective property of parabolas, page 599.

110. The graph of is a single point,

The plane intersects the double-napped cone at the vertices of the cones.

!1!2!3!4!5 1 2 3 4 5

!2!3!4!5

12345

x

y

!0, 0".x2 $ y2 " 0

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111.

For the upper half of the parabola,

y ! !6"x " 1# " 3.

y # 3 ! !6"x " 1#

"y # 3#2 ! 6"x " 1# 112.

For the lower half of the parabola,

y ! #1 # !2"x # 2#.

y " 1 ! #!2"x # 2#

"y " 1#2 ! 2"x # 2#

113.

Relative maximum:

Relative minimum: "0.67, 0.22#

"#0.67, 3.78#

f "x# ! 3x3 # 4x " 2 114.

Relative minimum: at x ! #0.75#1.13

f "x# ! 2x2 " 3x

115.

Relative minimum: "#0.79, 0.81#

f "x# ! x4 " 2x " 2 116.

Relative minimum: at 0.88

Relative maximum: 1.11 at #0.88

#3.11

f "x# ! x5 # 3x # 1

Section 9.2 Ellipses

! An ellipse is the set of all points the sum of whose distances from two distinct fixed points (foci)is constant.

! The standard equation of an ellipse with center and major and minor axes of lengths and is

(a) if the major axis is horizontal.

(b) if the major axis is vertical.

! where is the distance from the center to a focus.

! The eccentricity of an ellipse is e !ca

.

cc2 ! a2 # b2

"x # h#2

b2 ""y # k#2

a2 ! 1

"x # h#2

a2 ""y # k#2

b2 ! 1

2b2a"h, k#

"x, y#

Vocabulary Check

1. ellipse 2. major axis, center

3. minor axis 4. eccentricity

1.

Center:

Vertical major axis

Matches graph (b).

a ! 3, b ! 2

"0, 0#

x2

4"

y2

9! 1 2.

Center:

Horizontal major axis

Matches graph (c).

a ! 3, b ! 2

"0, 0#

x2

9"

y2

4! 1 3.

Center:

Vertical major axis

Matches graph (d).

a ! 5, b ! 2

"0, 0#

x2

4"

y2

25! 1

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Section 9.2 Ellipses 785

4.

Center:


Matches graph (f).

a ! 2, b ! 1

"0, 0#

x2

4" y2 ! 1 5.

Center:


Matches graph (a).

a ! 4, b ! 1

"2, #1#

"x # 2#2

16" "y " 1#2 ! 1 6.

Center:


Matches graph (e).

"#2, #2#

"x " 2#2

9"

"y " 2#2

4! 1

7.

Center:

Vertices:

Foci:

e !ca

!!55

8

"±!55, 0#"±8, 0#

c ! !64 # 9 ! !55

a ! 8, b ! 3,

"0, 0#

!2!4!10 2 4 10

!4!6!8

!10

2468

10

x

yx2

64"

y2

9! 1 8.

Center:

Vertices:

Foci:

e !ca

!!65

9

"0, ±!65#"0, ±9#

c ! !81 # 16 ! !65

a ! 9, b ! 4,

"0, 0#

!2!6!8!10 2 6 8 10

!4!6

!10

246

10

x

yx2

16"

y2

81! 1

9.

Center:

Vertices:

Foci:

x

y

!2!4 2 6 10!2

!4

!6

!8

2

4

6

e !ca

!35

"4, #1 ± 3#; "4, #4#, "4, 2#

"4, #1 ± 5#; "4, #6#, "4, 4#

a ! 5, b ! 4, c ! 3

"4, #1#

"x # 4#2

16"

"y " 1#2

25! 1 10.

Center:

Foci:

Vertices:

!1!3 !2!4!7 1

!2

1

2

3

4

6

x

ye !ca

!24

!12

"#3, 2 ± 4#; "#3, #2#, "#3, 6#

"#3, 2 ± 2#; "#3, 0#, "#3, 4#

a ! 4, b ! 2!3, c ! !16 # 12 ! 2

"#3, 2#

"x " 3#2

12"

"y # 2#2

16! 1

11.

Center:

Foci:

Vertices:

e !!5$23$2

!!53

%#5 #32

, 1& ! %#132

, 1&%#5 "32

, 1& ! %#72

, 1&,

%#5 "!52

, 1&, %#5 #!52

, 1&

a !32

, b ! 1, c !!94

# 1 !!52

"#5, 1#

1!1!3 !2!4!5!6!7

!2

!3

!4

4

2

3

1

x

y"x " 5#2

9$4" "y # 1#2 ! 1

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12.

Center:

Foci:

Vertices:

Eccentricity:!32

"#3, #4#, "#1, #4#

%#2 "!32

, #4&, %#2 #!32

, #4&"#2, #4#

a ! 1, b !12

, c ! !a2 # b2 !!32

–3 –2 –1 1

–5

–4

–3

–2

–1

x

y"x " 2#2 ""y " 4#2

1$4! 1

13. (a)

(b)

Center:

Vertices:

Foci:

e !ca

!4!2

6!

2!23

"±4!2, 0#"±6, 0#

"0, 0#

a ! 6, b ! 2, c ! !36 # 4 ! !32 ! 4!2

x2

36"

y2

4! 1

x2 " 9y2 ! 36 (c)

!8!10 6 8 10

!4!6!8

!10

468

10

x

y

14. (a)

(b)

Center:

Vertices:

Foci:

e !ca

!!15

4

"0, ±!15#"0, ±4#

"0, 0#

a ! 4, b ! 1, c ! !16 # 1 ! !15

x2 "y2

16! 1

16x2 " y2 ! 16 (c)

!2!3!4!5 2 3 4 5

!4!5

1

45

x

y

15. (a) (c)

(b)

Center:

Foci:

Vertices:

e !!53

"#2, 6#, "#2, 0#

"#2, 3 ± !5 #"#2, 3#

a ! 3, b ! 2, c ! !5

"x " 2#2

4"

"y # 3#2

9! 1

9"x2 " 4x " 4# " 4"y2 # 6y " 9# ! #36 " 36 " 36

1 2!1!3 !2!4!5!6

!2

4

6

2

3

x

y 9x2 " 4y2 " 36x # 24y " 36 ! 0

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16. (a) (c)

(b)

Center:

Foci:

Vertices:

e !2!5

6!

!53

"3, #5 ± 6#; "3, 1#, "3, #11#

"3, #5 ± 2!5#"3, #5#

a ! 6, b ! 4, c ! !20 ! 2!5

"x # 3#2

16"

"y " 5#2

36! 1

9"x # 3#2 " 4"y " 5#2 ! 144!2 2 4 10 128

2

!10

!12

y

x

9"x2 # 6x " 9# " 4"y2 " 10y " 25# ! #37 " 81 " 100

17. (a) (c)

(b)

Center:

Foci:

Vertices:

e !!2!3

!!63

%#32

, 52

± 2!3&%#

32

, 52

± 2!2&%#

32

, 52&

a ! 2!3, b ! 2, c ! 2!2

"x " 3

2#2

4"

"y # 52#2

12! 1

6%x "32&

2" 2%y #

52&

2! 24

6%x2 " 3x "94& " 2%y2 # 5y "

254 & ! #2 "

272

"252

2

2

4

!4

!2

!6x

y 6x2 " 2y2 " 18x # 10y " 2 ! 0

18. (a) (c)

(b)

Center:

Foci:

Vertices:

e !!32

%9, #52&, %#3, #

52&

%3 ± 3!3, #52&

%3, #52&

a ! 6, b ! 3, c ! !36 # 9 ! !27 ! 3!3

"x # 3#2

36"

"y " 52#2

9! 1

"x # 3#2 " 4%y "52&

2! 36

!3 !1!2!3!4

!6!7!8

21 3 4 5 6 9 10

21

3456

x

y "x2 # 6x " 9# " 4%y2 " 5y "254 & ! 2 " 9 " 25

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19. (a) (c)

(b)

Center:

Foci:

Vertices:

e !35

%94

, #1&, %#14

, #1&%7

4, #1&, %1

4, #1&

"1, #1#

a !54

, b ! 1, c !34

(x # 1)2

25$16" (y " 1)2 ! 1

16"x2 # 2x " 1# " 25"y2 " 2y " 1# ! #16 " 16 " 25

–2 –1 1 3

–3

–2

1

2

x

y 16x2 " 25y2 # 32x " 50y " 16 ! 0

20. (a) (c)

(b) Degenerate ellipse with center as the only point"2, 1#

9"x # 2#2 " 25"y # 1#2 ! 0

9"x2 # 4x " 4# " 25"y2 # 2y " 1# ! #61 " 36 " 25

1 2

1

2

x

y 9x2 " 25y2 # 36x # 50y " 61 ! 0

21. (a) (c)

(b)

Center:

Vertices:

Foci:

Eccentricity:ca

!!2!5

!!10

5

%12

± !2, #1&%1

2± !5, #1&

%12

, #1&a ! !5, b ! !3, c ! !5 # 3 ! !2

"x # 1

2#2

5"

"y " 1#2

3! 1

12%x #12&

2

" 20"y " 1#2 ! 60

12%x2 # 1 "14& " 20"y2 " 2y " 1# ! 37 " 3 " 20

x

y

!1!2!3 1 2 3

1

2

!2

!3

!4

12x2 " 20y2 # 12x " 40y # 37 ! 0

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23. Center:

Vertical major axis

x2

4"

y2

16! 1

a ! 4, b ! 2

"0, 0# 24. Vertices:

Endpoints of minor axis:

x2

4"

4y2

9! 1

x2

22 "y2

"3$2#2 ! 1

x2

a2 "y2

b2 ! 1

%0, ±32& ! b !

32

"±2, 0# ! a ! 2

25. Center:


x2

9"

y2

5! 1

c ! 2 ! b ! !9 # 4 ! !5a ! 3,


Foci:

Center:

y2

64"

x2

48! 1

"y # k#2

a2 ""x # h#2

b2 ! 1

"0, 0# ! "h, k#

b2 ! a2 # c2 ! 64 # 16 ! 48

"0, ±4# ! c ! 4

"0, ±8# ! a ! 8

27. Center:


x2

16"

y2

7! 1

a ! 4 ! b ! !16 # 9 ! !7

c ! 3

"0, 0# 28. Center:


x2

36"

y2

32! 1

a ! 6 ! b ! !36 # 4 ! !32 ! 4!2

c ! 2

"0, 0#

22. (a)

(c)

x

y

!1!2 1 2

!1

1

3

"x " 2

3#2

14

""y # 2#2

1! 1

36%x "23&

2

" 9"y # 2#2 ! 9

36%x2 "43

x "49& " 9"y2 # 4y " 4# ! #43 " 16 " 36

36x2 " 9y2 " 48x # 36y " 43 ! 0 (b)

Center:

Vertices:

Foci:

Eccentricity:ca

!!32

%#23

, 2 ±!32 &

%#23

, 2 ± 1& ! %#23

, 1&, %#23

, 3&%#

23

, 2&

c !!1 #14

!!32

b !12

,a ! 1,

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29. Vertices:

Center:

Vertical major axis

Point:

21x2

400"

y2

25! 1

x2

400$21"

y2

25! 1

40021

! b2

400 ! 21b2

16b2 ! 1 #

425

!2125

42

b2 "22

25! 1

"4, 2#

x2

b2 "y2

25! 1

"x # h#2

b2 ""y # k#2

a2 ! 1

"0, 0#

"0, ±5# ! a ! 5 30. Vertical major axis

Passes through: and

x2

4"

y2

16! 1

x2

b2 "y2

a2 ! 1

a ! 4, b ! 2

"2, 0#"0, 4#

31. Center:

Vertical major axis

"x # 2#2

1"

"y # 3#2

9! 1

"x # h#2

b2 ""y # k#2

a2 ! 1

a ! 3, b ! 1


Center:


"x # 2#2

4"

"y " 1#2

1! 1

"x # h#2

a2 ""y # k#2

b2 ! 1

"2, 0#, "2, #2# ! b ! 1

"2, #1# ! h ! 2, k ! #1

"0, #1#, "4, #1# ! a ! 2

33. Center:


"x # 4#2

16"

" y # 2#2

1! 1

a ! 4, b ! 1 ! c ! !16 # 1 ! !15

"4, 2# 34. Center:


"x # 2#2

9"

y2

5! 1

c ! 2, a ! 3 ! b2 ! a2 # c2 ! 9 # 4 ! 5

"2, 0#

35. Center:

Vertical major axis

x2

308"

" y # 4#2

324! 1

c ! 4, a ! 18 ! b2 ! a2 # c2 ! 324 # 16 ! 308

"0, 4#

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36. Center:

Vertex:

Minor axis length:

"x # 2#2 "4"y " 1#2

9! 1

"x # 2#2

1"

"y " 1#2

"3$2#2 ! 1

"x # h#2

b2 ""y # k#2

a2 ! 1

2 ! b ! 1

%2, 12& ! a !

32

"2, #1# ! h ! 2, k ! #1 37. Vertices:

Center:

Minor axis of length

Vertical major axis

"x # 3#2

9"

"y # 5#2

16! 1

"x # h#2

b2 ""y # k#2

a2 ! 1

6 ! b ! 3

"3, 5#

"3, 1#, "3, 9# ! a ! 4

38. Center:

Foci:

"x # 3#2

36"

"y # 2#2

32! 1

"x # h#2

a2 ""y # k#2

b2 ! 1

b2 ! a2 # c2 ! 36 # 4 ! 32

"1, 2#, "5, 2# ! c ! 2, a ! 6

a ! 3c

"3, 2# ! "h, k# 39. Center:

Vertices:


x2

16"

"y # 4#2

12! 1

"x # h#2

a2 ""y # k#2

b2 ! 1

22 ! 42 # b2 ! b2 ! 12

a ! 2c ! 4 ! 2c ! c ! 2

"#4, 4#, "4, 4# ! a ! 4

"0, 4#

43.

e !ca

!2!2

3

a ! 3, b ! 1, c ! !9 # 1 ! 2!2

"x # 5#2

9"

" y " 2#2

1! 1

"x # 5#2 " 9" y " 2#2 ! 9

"x2 # 10x " 25# " 9" y2 " 4y " 4# ! #52 " 25 " 36

x2 " 9y2 # 10x " 36y " 52 ! 0

40. Vertices:


Center:

"x # 5#2

25"

"y # 6#2

36! 1

"x # h#2

b2 ""y # k#2

a2 ! 1

"5, 6# ! h ! 5, k ! 6

"0, 6#, "10, 6# ! b ! 5

"5, 0#, "5, 12# ! a ! 641.

e !ca

!!53

c ! !9 # 4 ! !5

a ! 3, b ! 2,

x2

4"

y2

9! 1 42.

e !ca

!!11

6

c ! !36 # 25 ! !11

a ! 6, b ! 5,

x2

25"

y2

36! 1

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44.

e !ca

!12

a ! 2, b ! !3, c ! !4 # 3 ! 1

"x # 1#2

3"

" y " 3#2

4! 1

4"x # 1#2 " 3" y " 3#2 ! 12

4"x2 # 2x " 1# " 3" y2 " 6y " 9# ! #19 " 4 " 27

4x2 " 3y2 # 8x " 18y " 19 ! 0

45. Vertices:

Eccentricity:

Center:


x2

25"

y2

9! 1

"0, 0#

b2 ! a2 # c2 ! 25 # 16 ! 9

45

!ca

! c !45

a ! 4

"±5, 0# ! a ! 5 46. Vertices:

Eccentricity:

x2

48"

y2

64! 1

x2

b2 "y2

a2 ! 1

b2 ! a2 # c2 ! 64 # 16 ! 48

c ! 4

12

!c8

e !12

!ca

h ! 0, k ! 0"0, ±8# ! a ! 8,

47. (a)

!20!40 20 40

!20

20

60

80

(!50, 0) (50, 0)

(0, 40)

x

y (b) Vertices:

Height at center:


x2

2500"

y2

1600! 1, y " 0

x2

a2 "y2

b2 ! 1

40 ! b ! 40

"±50, 0# ! a ! 50 (c) For

The height five feet from the edge of the tunnel is approximately 17.44 feet.

y ' 17.44

y2 ! 304

y2 ! 1600%1 #452

2500&

452

2500"

y2

1600! 1.x ! 45,

48. (a)

!4!8!12!20 4 8 12 16 20

!8!12!16!20

48

1620

x

y

(0, 12)

(!16, 0) (16, 0)

(b)

x2

256"

y2

144! 1, y " 0

a ! 16, b ! 12 (c) When

Hence, the truck will be ableto drive through without crossing the center line.

y ' 9.4 > 9.

y2 ! 144!1 #102

256

x ! 10,

49. Let be the equation of the ellipse. Then and

Thus, the tacks are placed

at The string has a length of 2a ! 6 feet."±!5, 0#.c2 ! a2 # b2 ! 9 # 4 ! 5.a ! 3 !

b ! 2x2

a2 "y2

b2 ! 1

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50.

Distance between foci: feet2"4.7# ' 85.4

a !972

, b ! 23, c !!%972 &2

# "23#2 ' 4.7

%or x2

232 "y2

"97$2#2 ! 1&x2

"97$2#2 "y2

232 ! 1

x

y

!20 20 40

!40

40

51.

Length of major axis: 2a ! 2"20# ! 40 units

a ! 20

$a"10# ! 200

$a"10# ! 2$"10#2

$ab ! 2$r2

Area of ellipse ! 2"area of circle#

52. Center:

Ellipse:x2

321.84"

y2

19.02! 1

c2 ! a2 # b2 ! b2 ! a2 # c2 ' 19.02

e !ca

! 0.97 !c

17.94 ! c ! 17.4018

2a ! 35.88 ! a ! 17.94 ! a2 ' 321.84

"0, 0#, e ! 0.97 53.

x2

4.8841"

y2

1.3872! 1

b2 ! a2 # c2 ! b2 ! 1.3872

2a ! 4.42 ! a ! 2.21 ! c ! 1.87

a # c ! 0.34

a " c ! 4.08

54.

e !ca

' 0.0516

! 359.5

c ! 7325 # 6965.5

a ! 6965.5

x

b

ac!a

!b

y

a ! c

a + c

2a ! 13,931

a # c ! 228 " 6378 ! 6606

a " c ! 947 " 6378 ! 7325 55. For we have

When

! 2y !2b2

a.

! y2 !b4

a2

c2

a2 "y2

b2 ! 1 ! y2 ! b2%1 #a2 # b2

a2 &x ! c,

c2 ! a2 # b2.x2

a2 "y2

b2 ! 1,

56.

Points on the ellipse:

Length of latus recta:

Additional points: %!3, ±12&, %#!3, ±

12&

2b2

a! 1

"±2, 0#, "0, ±1#

a ! 2, b ! 1, c ! !3

!1

!2

2

1x

(

(

(

(

, !

,

!1

1

1

1

2

2

2

2

)

)

)

)

! 3

! 3

3,

3,

yx2

4"

y2

1! 1 57.



Additional points: %±94

, #!7&, %±94

, !7&

2b2

a!

2"3#2

4!

92

"±3, 0#, "0, ±4#

a ! 4, b ! 3, c ! !7

x

!

!

9

9 9

94

4 4

4,

, ,

7

7 7! !

, 7(

( (

()

) )

)

y

!2!4 2 4

!2

2

x2

9"

y2

16! 1

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58.



Additional points: %±43

, #!5&, %±43

, !5&

2b2

a!

2 % 22

3!

83

"±2, 0#, "0, ±3#

x2

4"

y2

9! 1

!1!3

!2

2

1 3x

( (

((

, ,! 5 ! 5

5, 5 ,

4

4

4

4

3

3

3

3

) )

))

!

!

y9x2 " 4y2 ! 36 59.



Additional points: %±3!5

5, #!2&, %±

3!55

, !2&

2b2

a!

2 % 3!5

!6!5

5

"±!3, 0#, "0, ±!5#

c ! !2

a ! !5, b ! !3,

x2

3"

y2

5! 1

!4 !2 2 4

!4

4

x

(

((

( , 2

, 2!, 2!

, 2 3 5

3 53 5

3 55

55

5 )

))

)

!

!

y 5x2 " 3y2 ! 15

60. Answers will vary. 61. True. If then the ellipse is elongated, notcircular.

e ' 1

62. True. The ellipse is inside the circle. 63. (a) The length of the string is

(b) The path is an ellipse because the sum of thedistances from the two thumbtacks is alwaysthe length of the string, that is, it is constant.

2a.

64. (a)

(b)

by the Quadratic Formula

Since we choose

x2

196"

y2

36! 1

x2

142 "y2

62 ! 1

a ! 14 and b ! 6.a > b,

b ! 64 ORb ! 14

a ' 14 or a ' 6

$a2 # 20$a " 264 ! 0

264 ! $a"20 # a#

A ! $ab ! $a"20 # a#

a " b ! 20 ! b ! 20 # a (c)

(d)

The area is maximum when and it is a circle.

a ! b ! 10

00 24

360

8 9 10 11 12 13

301.6 311.0 314.2 311.0 301.6 285.9A

a

65. Center:

Foci:


"x # 6#2

324"

" y # 2#2

308! 1

b2 ! a2 # b2 ! b ! !182 # 16 ! !308

"a " c# " "a # c# ! 2a ! 36 ! a ! 18

"2, 2#, "10, 2# ! c ! 4

"6, 2# 66.

The sum of the distancesfrom any point on theellipse to the two foci isconstant. Using the vertex

you have

From the figure,

2!b2 " c2 ! 2a ! a2 ! b2 " c2.

"a " c# " "a # c# ! 2a.

"a, 0#,

x

b

b

acc

!c!a

!b

y

b2 + c2

x2

a2 "y2

b2 ! 1

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Section 9.3 Hyperbolas 795

67. Arithmetic: d ! "11 68. Geometric: r ! 12 69. Geometric: r ! 2 70. Arithmetic: d ! 1

71. !6

n!0 3n ! 1093 72. !

6

n!0 ""3#n ! 547 73. !

10

n!1 4"3

4#n"1$ 15.099 74. !10

n!0 5"4

3#n $ 340.155

Section 9.3 Hyperbolas

! A hyperbola is the set of all points the difference of whose distances from two distinct fixed points(foci) is constant.

! The standard equation of a hyperbola with center and transverse and conjugate axes of lengths and is:

(a) if the transverse axis is horizontal.

(b) if the transverse axis is vertical.

! where is the distance from the center to a focus.

! The asymptotes of a hyperbola are:

(a) if the transverse axis is horizontal.

(b) the transverse axis is vertical.

! The eccentricity of a hyperbola is

! To classify a nondegenerate conic from its general equation (a) If then it is a circle.(b) If but not both), then it is a parabola.(c) If then it is an ellipse.(d) If then it is a hyperbola.AC < 0,

AC > 0, AC ! 0 (A ! 0 or C ! 0, A ! C (A # 0, C # 0),

Ax2 $ Cy2 $ Dx $ Ey $ F ! 0:

e !ca

.

y ! k ±ab

"x " h#

y ! k ±ba

"x " h#

cc2 ! a2 $ b2

"y " k#2

a2 ""x " h#2

b2 ! 1

"x " h#2

a2 ""y " k#2

b2 ! 1

2b2a"h, k#

"x, y#

Vocabulary Check1. hyperbola 2. branches 3. transverse axis, center

4. asymptotes 5. Ax2 $ Cy2 $ Dx $ Ey $ F ! 0

1. Center:

Vertical transverse axis

Matches graph (b).

a ! 3, b ! 5, c ! %34

"0, 0# 2. Center:


Matches graph (c).

a ! 5, b ! 3

"0, 0#

3. Center:

Horizontal transverse axis

Matches graph (a).

a ! 4, b ! 2

"1, 0# 4. Center:


Matches graph (d).

a ! 4, b ! 3

""1, 2#©H

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5.

Center:

Vertices:

Foci:

Asymptotes: y ! ±x

"±%2, 0#"±1, 0#

"0, 0#

a ! 1, b ! 1, c ! %2

–2 2

–2

–1

1

2

x

yx2 " y2 ! 1 6.

Center:

Vertices:

Foci:

Asymptotes: y ! ±ba

x ! ±53

x

"±%34, 0#"±3, 0#

c ! %32 $ 52 ! %34

a ! 3, b ! 5,

"0, 0#

!4

!6!8

!10

!6!8 2 4 6 8 10

468

10

x

yx2

9"

y2

25! 1

7.

Center:

Vertices:

Foci:

Asymptotes: y ! ±12

x

"0, ±%5 #"0, ±1#

"0, 0#

a ! 1, b ! 2, c ! %5

–3 –2 2 3

–3

–2

2

3

y

x

y2

1"

x2

4! 1 8.

Center:

Vertices:

Foci:

Asymptotes: y ! ±3x

"0, ±%10#"0, ±3#

"0, 0#

c ! %32 $ 12 ! %10

a ! 3, b ! 1,

–6 –4 –2 2 4 6

–6

6

x

yy2

9"

x2

1! 1

9.

Center:

Vertices:

Foci:

Asymptotes:

y ! ±ab

x ! ±59

x

"0, ±%106 #"0, ±5#

x

y

!6!9 6 9 12 15!3

!9!12!15

3

91215

"0, 0#

a ! 5, b ! 9, c ! %a2 $ b2 ! %106

y2

25"

x2

81! 1 10.

Center:

Vertices:

Foci:

Asymptotes: y ! ±13

x

"±2%10, 0#"±6, 0#

"0, 0#

c ! %36 $ 4 ! 2%10

a ! 6, b ! 2,

–12 12

–12

–8

–4

4

8

12

x

yx2

36"

y2

4! 1

11.

Center:

Vertices:

Foci:

Asymptotes: y ! "2 ±12

"x " 1#

"1 ± %5, "2#""1, "2#, "3, "2#

"1, "2#

a ! 2, b ! 1, c ! %5

1 2 3

–5

–4

1

2

3

x

y"x " 1#2

4"

"y $ 2#2

1! 1 12.

Center:

Vertices:

Foci:

Asymptotes:

y ! 2 ±512

"x $ 3#

""16, 2#, "10, 2#

""15, 2#, "9, 2#

a ! 12, b ! 5, c ! 13

""3, 2#5

10

15

!5 5!5

!10

!15

!20

x

y"x $ 3#2

144"

"y " 2#2

25! 1

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13.

Center:

Vertices:

Foci:

Asymptotes:

x

y

!1!2 1 2 3 4!1

!2

!3

!5

y ! "5 ±23

"x " 1#

y ! k ±ab

"x " h#

&1, "5 ±%13

6 '&1, "5 ±

13': &1, "

163 ', &1, "

143 '

"1, "5#

a !13

, b !12

, c !%19

$14

!%13

6

"y $ 5#2

1(9"

"x " 1#2

1(4! 1 14.

Center:

Vertices:

Foci:

Asymptotes:

–3 –1

–1

1

2

3

x

y

y ! 1 ±1(21(4

"x $ 3# ! 1 ± 2"x $ 3#

&"3, 1 ±%54 '

&"3, 12', &"3,

32'

a !12

, b !14

, c !%14

$116

!%54

""3, 1#

"y " 1#2

1(4"

"x $ 3#2

1(16! 1

15. (a)

(b) Center:

Vertices:

Foci:

Asymptotes:

(c)

!2!4!5 2 4 5

!2!3!4!5

12345

x

y

y ! ±ba

x ! ±23

x

"±%13, 0#"±3, 0#

a ! 3, b ! 2, c ! %9 $ 4 ! %13

"0, 0#

x2

9"

y2

4! 1

4x2 " 9y2 ! 36 16. (a)

(b) Center:

Vertices:

Foci:

Asymptotes:

(c)

!4!6!8 4 6 8

!6

!8

6

8

4

x

y

y ! ±ba

x ! ±52

x

"±%29, 0#"±2, 0#

a ! 2, b ! 5, c ! %4 $ 25 ! %29

"0, 0#

x2

4"

y2

25! 1

25x2 " 4y2 ! 100

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17. (a)

(b)

Center:

Vertices:

Foci:

! ±%63

x

Asymptotes: y ! ±%23

x

"±%5, 0#"±%3, 0#

"0, 0#

a ! %3, b ! %2, c ! %5

x2

3"

y2

2! 1

2x2 " 3y2 ! 6 (c) To use a graphing calculator, solve first for

y4 ! "%23

x

y3 !%23

x

y2 ! "%2x2 " 63

!3!4 3 4

!3

!2

!4

1

2

3

4

x

yy1 !%2x2 " 63

y2 !2x2 " 6

3

y.

) Asymptotes

) Hyperbola

18. (a)

(b)

Center:

Vertices:

Foci:

Asymptotes: y ! ±%3%6

x ! ±%22

x

"0, ±3#

"0, ±%3#"0, 0#

a ! %3, b ! %6, c ! 3

y 2

3"

x 2

6! 1

6y 2 " 3x 2 ! 18 (c)

x

y

!4 !3 !2 432!1

!3

!4

1

3

4

19. (a)

(b)

Center:

Vertices:

Foci:

Asymptotes: y ! "3 ± 3"x " 2#

"2 ± %10, "3#"1, "3#, "3, "3#

"2, "3#

a ! 1, b ! 3, c ! %10

"x " 2#2

1"

"y $ 3#2

9! 1

9"x2 " 4x $ 4# " "y2 $ 6y $ 9# ! "18 $ 36 " 9

9x2 " y2 " 36x " 6y $ 18 ! 0 (c)

–6 –4 –2 2 4 6 8

–8

–6

–4

2

x

y

20. (a)

x2

36"

"y " 2#2

4! 1

x2 " 9"y " 2#2 ! 36

x2 " 9" y2 " 4y $ 4# ! 72 " 36

x2 " 9y2 $ 36y " 72 ! 0 (b)

Center:

Vertices:

Foci:

Asymptotes: y ! 2 ±13

x

"±2%10, 2#"±6, 2#

"0, 2#

c ! %36 $ 4 ! 2%10

a ! 6, b ! 2, (c)

–8 –4 4 8

–12

–8

–4

4

8

12

x

y

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21. (a)

(b) Degenerate hyperbola is two lines intersecting at""1, "3#.

y $ 3 ! ±13"x $ 1#

"x $ 1#2 " 9"y $ 3#2 ! 0

"x2 $ 2x $ 1# " 9"y2 $ 6y $ 9# ! 80 $ 1 " 81

x2 " 9y2 $ 2x " 54y " 80 ! 0 (c)

–4 –2 2

–6

–4

–2

2

4

x

y

22. (a)

(b) Degenerate hyperbola is two intersecting lines at "1, "2#.

y $ 2 ! ±14"x " 1#

16"y $ 2#2 " "x " 1# ! 0

16"y2 $ 4y $ 4# " "x2 " 2x $ 1# ! "63 $ 64 " 1

16y2 " x2 $ 2x $ 64y $ 63 ! 0 (c)

–1 1 2 3

–4

–3

–2

–1

x

y

23. (a)

(b)

Center:

Vertices:

Foci:

Asymptotes:

(c) To use a graphing calculator, solve for first.

y4 ! "3 "13

"x " 1#

y3 ! "3 $13

"x " 1#

y2 ! "3 "13%18 $ "x " 1#2

y1 ! "3 $13%18 $ "x " 1#2

y ! "3 ± %18 $ "x " 1#2

9

9"y $ 3#2 ! 18 $ "x " 1#2

x

y

2

!6

!8

!10

2

4

y

y ! "3 ±13

"x " 1#

"1, "3 ± 2%5 #"1, "3 ± %2 #

"1, "3#

a ! %2, b ! 3%2, c ! 2%5

"y $ 3#2

2"

"x " 1#2

18! 1

9"y2 $ 6y $ 9# " "x2 " 2x $ 1# ! "62 " 1 $ 81

9y2 " x2 $ 2x $ 54y $ 62 ! 0

) Asymptotes

) Hyperbola

©H

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24. (a)

(b)

Center:

Vertices:

Foci:

Asymptotes: y ! 5 ± 3"x $ 3#

&"3 ±%10

3, 5'

&"3 ±13

, 5'""3, 5#

a !13

, b ! 1, c !%10

3

"x $ 3#2

1(9"

"y " 5#2

1! 1

9"x2 $ 6x $ 9# " "y2 " 10y $ 25 # ! "55 $ 81 " 25

9x2 " y2 $ 54x $ 10y $ 55 ! 0 (c)

x

y

!2!3!4!6!7 1

2

4

6

8

10

14

28. Vertices:

Asymptotes:

Center:

y2

9" x2 ! 1

"y " k#2

a2 ""x " h#2

b2 ! 1

"0, 0# ! "h, k#

y ! ±3x ! ab

! 3, b ! 1

"0, ±3# ! a ! 3 29. Foci:

Asymptotes:

Center:

17y2

1024"

17x2

64! 1

y2

1024(17"

x2

64(17! 1

"y " k#2

a2 ""x " h#2

b2 ! 1

6417

! b2 ! a2 !102417

c2 ! a2 $ b2 ! 64 ! 16b2 $ b2

"0, 0# ! "h, k#

y ! ±4x ! ab

! 4 ! a ! 4b

"0, ±8# ! c ! 8

25. Vertices:

Foci:

Center:

y2

4"

x2

12! 1

"y " k#2

a2 ""x " h#2

b2 ! 1

"0, 0# ! "h, k#

b2 ! c2 " a2 ! 16 " 4 ! 12

"0, ±4# ! c ! 4

"0, ±2# ! a ! 2 26. Vertices:

Foci:

x2

9"

y2

27! 1

x2

a2 "y2

b2 ! 1

b2 ! c2 " a2 ! 36 " 9 ! 27

"±6, 0# ! c ! 6

"±3, 0# ! a ! 3 27. Vertices:

Asymptotes:

Center:

x2

1"

y2

25! 1

"0, 0#

! b ! 5

y ! ±5x ! ba

! 5

"±1, 0# ! a ! 1

©H

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30. Foci:

Asymptotes:

x2

64"

y2

36! 1

x2

a2 "y2

b2 ! 1

b ! 3"2# ! 6a ! 4"2# ! 8,

2 ! m

100 ! 25m2

c2 ! a2 $ b2 ! 100 ! "3m#2 $ "4m#2

y ! ±34

x ! ba

!3m4m

"±10, 0# ! c ! 10 31. Vertices:

Foci:

Center:

"x " 4#2

4"

y2

12! 1

"x " h#2

a2 ""y " k#2

b2 ! 1

"4, 0# ! "h, k#

b2 ! c2 " a2 ! 16 " 4 ! 12

"0, 0#, "8, 0# ! c ! 4

"2, 0#, "6, 0# ! a ! 2

32. Vertices:

Center:

Foci:

y2

9"

"x " 2#2

16! 1

"y " k#2

a2 ""x " h#2

b2 ! 1

b2 ! c2 " a2 ! 25 " 9 ! 16

"2, "5# ! c ! 5"2, 5#,

"2, 0#

"2, "3# ! a ! 3"2, 3#, 33. Vertices:

Foci:

Center:

"y " 5#2

16"

"x " 4#2

9! 1

"y " k#2

a2 ""x " h#2

b2 ! 1

"4, 5# ! "h, k#

b2 ! c2 " a2 ! 25 " 16 ! 9

"4, 0#, "4, 10# ! c ! 5

"4, 1#, "4, 9# ! a ! 4

34. Vertices:

Center:

Foci:

x2

4"

"y " 1#2

5! 1

"x " h#2

a2 ""y " k#2

b2 ! 1

b2 ! c2 " a2 ! 9 " 4 ! 5

"3, 1# ! c ! 3""3, 1#,

"0, 1#

"2, 1# ! a ! 2""2, 1#, 35. Vertices:

Solution point:

Center:

y2

9"

"x " 2#2

9(4! 1

!9""2#2

25 " 9!

3616

!94

b2 !9"x " 2#2

y2 " 9

y2

9"

"x " 2#2

b2 ! 1 !

"y " k#2

a2 ""x " h#2

b2 ! 1

"2, 0# ! "h, k#

"0, 5#

"2, 3#, "2, "3# ! a ! 3

©H

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36. Center:

Solution point:

x2

4"

"y " 1#2

12(7! 1

b2 !3621

!127

9b2 !

214

254

"9b2 ! 1

"5, 4#

x2

4"

"y " 1#2

b2 ! 1

"0, 1#, a ! 2 37. Vertices:

Center:

Passes through

"y " 2#2

4"

x2

4! 1

b2 ! 4 ! b ! 2

94

" 1 !5b2

""1 " 2#2

4"

5b2 ! 1

"%5, "1#

"y " 2#2

4"

x2

b2 ! 1

"0, 2#, a ! 2

"0, 4#, "0, 0#

41. Vertices:

Asymptotes:

Center:

"x " 3#2

9"

"y " 2#2

4! 1

"x " h#2

a2 ""y " k#2

b2 ! 1

"3, 2# ! "h, k#

ba

!23

! b ! 2

y !23

x, y ! 4 "23

x

"0, 2#, "6, 2# ! a ! 3 42. Vertices:

Asymptotes:

Center:

"y " 2#2

4"

"x " 3#2

9! 1

"y " k#2

a2 ""x " h#2

b2 ! 1

"3, 2# ! "h, k#

ab

!23

! b ! 3

y !23

x, y ! 4 "23

x

(3, 0#, "3, 4# ! a ! 2

38. Center:

Solution point:

y2

4"

"x " 1#2

4! 1

1b2 !

14

! b ! 2

54

"1b2 ! 1

"0, %5#

y2

4"

"x " 1#2

b2 ! 1

"1, 0#, a ! 2 39. Vertices:

Center:

Asymptotes:

"x " 2#2

1"

"y " 2#2

1! 1

ba

! 1 ! b ! 1

y ! x, y ! 4 " x

"2, 2#

"1, 2#, "3, 2# ! a ! 1

40. Center:

Asymptotes:

"y $ 3#2

9"

"x " 3#2

9! 1

1 !ab

!3b

! b ! 3

y ! x " 6, y ! "x

"3, "3#, a ! 3

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43. Friend’s location

Your location

Location of lightning strike

x2

98,010,000"

y2

13,503,600

b2 ! c2 " a2 ! 13,503,600

c ! 10,560, a !19,800

2! 9900 ! a2 ! 98,010,000

x2

a2 "y2

b2 ! 1

"1100#"18# ! 19,800

P"x, y#:

"10,560, 0#F2:

!20,000 20,000

!10,000

10,000

x

y

P

F

(10,560, 0)(!10,560, 0)

F1 2Friend You

""10,560, 0#F1:

44. The explosion occurred on the vertical line throughand

Hence,

The explosion occurred on the hyperbola

Letting

"3300, "2750#

y2 ! b2&x2

a2 " 1' ! "33002 " 22002#&33002

22002 " 1' ! y ! "2750.

x ! 3300,

x2

a2 "y2

b2 ! 1.

b2 ! c2 " a2.

c ! 3300

a ! 2200

2a ! 4400

d2 " d1 ! 4"1100# ! 4400

"3300, 0#."3300, 1100# (3300, 1100)

(3300, 0)( 3300, 0)!

d1d2

1000

2000

3000

4000

ax

y

!4000

!4000

45. (a)

is on the curve, so

x2

1"

y2

27! 1, "9 " y " 9

! b2 !813

! b ! 3%3.

41

"81b2 ! 1 ! 81

b2 ! 3

a ! 1; "2, 9#

x2

a2 "y2

b2 ! 1 (b) Because each unit is foot, 4 inches is of a unit. The base is 9 units from the origin, so

When

So the width is units, or22.68 inches, or 1.88998 feet.

2x $ 3.779956

x2 ! 1 $"25(3#2

27 ! x $ 1.88998.

y !253

,

y ! 9 "23

! 813

.

23

12

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46. Foci:

Center:

(a)

(b) 150 " 93 ! 57 miles

x $ 110.3 miles

x2 ! 932&1 $752

13,851' $ 12,161.43

x2

932 "y2

13,851! 1

b2 ! c2 " a2 ! 1502 " 932 ! 13,851

! 186 ! 2a ! 186 ! a ! 93

d2 " d1 ! "186,000#"0.001#

d1d2

15075

75

150

(150, 0)x

y

!75!150

(!150, 0)

(x, 75)

"0, 0#

"±150, 0# ! c ! 150

(c) Bay to Station 1: 30 miles

Bay to Station 2: 270 miles

(d) In this case,

and The hyperbola is

For and

Position: "144.2, 60#

x $ 144.2.y ! 60, x2 ! 20,800

x2

1202 "y2

902 ! 1.

b2 ! c2 " a2 ! 8100.

d2 " d1 ! 186,000"0.00129# $ 239.94 ! a $ 120

"270 " 30#186,000

$ 0.00129 second

47. Center:

Focus:

Since and we choose The vertex is approximate at [Note: By the Quadratic Formula, the exact value of is ]a ! 12"%5 " 1#.a

"14.83, 0#.a ! 14.83.c ! 24,a < c

a $ ±38.83 or a $ ±14.83

a4 " 1728a2 $ 331,776 ! 0

576"576 " a2# " 576a2 ! a2"576 " a2#

576a2 "

576576 " a2 ! 1

242

a2 "242

576 " a2 ! 1

x2

a2 "y2

576 " a2 ! 1

b2 ! c2 " a2 ! 242 " a2 ! 576 " a2

"24, 0#

"0, 0#

48.

The camera is units from the mirror.5 $ %41

a ! 5, b ! 4, c ! %25 $ 16 ! %41

x2

25"

y2

16! 1 49.

EllipseAC ! 36 > 0,

A ! 9, C ! 4

9x2 $ 4y2 " 18x $ 16y " 119 ! 0 ©H

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50.

CircleA ! C ! 1,

x2 $ y2 " 4x " 6y " 23 ! 0 51.

HyperbolaAC ! 16""9# < 0,

A ! 16, C ! "9

16x2 " 9y2 $ 32x $ 54y " 209 ! 0

52.

ParabolaAC ! 0,

A ! 1, C ! 0

x2 $ 4x " 8y $ 20 ! 0 53.

ParabolaAC ! 0,

C ! 1, A ! 0

y2 $ 12x $ 4y $ 28 ! 0

54.

EllipseAC ! 100 > 0,

A ! 4, C ! 25

4x2 $ 25y2 $ 16x $ 250y $ 541 ! 0 55.

CircleA ! C ! 1,

x2 $ y2 $ 2x " 6y ! 0

56.

HyperbolaAC < 0,

A ! "1, C ! 1

y2 " x2 $ 2x " 6y " 8 ! 0 57.

AC ! 0 ! Parabola

E ! "2, F ! 7A ! 1, C ! 0, D ! "6,

x2 " 6x " 2y $ 7 ! 0

58.

AC ! 9"4# ! 36 > 0 ! Ellipse

A ! 9, C ! 4

9x2 $ 4y2 " 90x $ 8y $ 228 ! 0 59. True. e !ca

!%a2 $ b2

a

60. False. because it is in the denominator.b # 0

61. False. For example,

is the graph of two intersecting lines.

"x " 1#2 " " y " 1#2 ! 0

x2 " y2 " 2x $ 2y ! 0

62. True. The asymptotes are

If they intersect at right angles, then

ba

!"1

""b(a# !ab

! a ! b.

y ! ±ba

x.

63. Let be such that the difference of the distances from and is (again only deriving one of the forms).

Let Then a2b2 ! b2x2 " a2y2 ! 1 !x2

a2 "y2

b2.b2 ! c2 " a2.

a2"c2 " a2# ! "c2 " a2#x2 " a2y2

a2"x2 " 2cx $ c2 $ y2# ! c2x2 " 2a2cx $ a4

a%"x " c#2 $ y2 ! cx " a2

4a%"x " c#2 $ y2 ! 4cx " 4a2

4a2 $ 4a%"x " c#2 $ y2 $ "x " c#2 $ y2 ! "x $ c#2 $ y2

2a $ %"x " c#2 $ y2 ! %"x $ c#2 $ y2

2a ! *%"x $ c#2 $ y2 " %"x " c# $ y2*2a

""c, 0#"c, 0#"x, y#

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64. Answers will vary. See Example 3. 65.

At the point

*d2 " d1* ! *"a $ c# " "c " a#* ! 2a.

"a, 0#,*d2 " d1* ! constant by definition of hyperbola

66. Center:


Foci at and

"x " 6#2

9"

" y " 2#2

7! 1

b2 ! c2 " a2 ! 16 " 9 ! 7

"c $ a# " "c " a# ! 6 ! a ! 3

"10, 2# ! c ! 4."2, 2#

"6, 2#

67. At the point the difference of the distances to the foci is Let be a point on the hyperbola.

Thus, as desired.c2 " a2 ! b2,

1 !x2

a2 "y2

c2 " a2

a2"c2 " a2# ! "c2 " a2#x2 " a2y2

a2"x2 " 2cx $ c2 $ y2# ! c2x2 " 2a2cx $ a4

a%"x " c#2 $ y2 ! cx " a2

4a%"x " c#2 $ y2 ! 4cx " 4a2

4a2 $ 4a%"x " c#2 $ y2 $ "x " c#2 $ y2 ! "x $ c#2 $ y2

2a $ %"x " c#2 $ y2 ! %"x $ c#2 $ y2

2a ! %"x $ c#2 $ y2 " %"x " c#2 $ y2

"x, y#"c $ a# " "c " a# ! 2a."±c, 0#"a, 0#,

68. If then by completing the square you obtain a circle.

If and then is a parabola (complete the square).Same for and

If then both and are positive (or both negative). By completing the squareyou obtain an ellipse.

If then and have opposite signs. You obtain a hyperbola.CAAC < 0,

CAAC > 0,

C ! 0.A # 0Cy2 $ Dx $ Ey $ F ! 0C # 0,A ! 0

A ! C # 0,

69. "x3 " 3x2# " "6 " 2x " 4x2# ! x3 $ x2 $ 2x " 6 70.

! 3x2 $ 232 x " 2

"3x " 12#"x $ 4# ! 3x2 $ 12x " 1

2x " 2

71.

x3 " 3x $ 4x $ 2

! x2 " 2x $ 1 $2

x $ 2

"2 1

1

0"2

"2

"34

1

4"2

2

72.

! x2 $ 2xy $ y2 $ 6x $ 6y $ 9

+"x $ y# $ 3,2 ! "x $ y#2 $ 6"x $ y# $ 9

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Section 9.4 Rotation and Systems of Quadratic Equations 807

Section 9.4 Rotation and Systems of Quadratic Equations

! The general second-degree equation can be rewritten by rotating the coordinate axes through the angle

where

!

! The graph of the nondegenerate equation is:

(a) An ellipse or circle if

(b) A parabola if

(c) A hyperbola if B2 ! 4AC > 0.

B2 ! 4AC " 0.

B2 ! 4AC < 0.

Ax2 # Bxy # Cy2 # Dx # Ey # F " 0

y " x$ sin % # y$ cos %x " x$ cos % ! y$sin %

cot 2% " !A ! C"#B.%A$!x$ "2 # C$!y$ "2 # D$x$ # E$y$ # F$ " 0

Ax2 # Bxy # Cy2 # Dx # Ey # F " 0

Vocabulary Check1. rotation, axes 2. invariant under rotation 3. discriminant

1. Point:

Thus, !x$, y$ " " !3, 0".

3 " x$0 " y$

3 " x$ sin 90& # y$ cos 90&0 " x$ cos 90& ! y$ sin 90&

y " x$ sin % # y$ cos %x " x$ cos % ! y$ sin %

!0, 3"% " 90&;

2. Point:

Adding,

Subtracting,

Thus, !x$, y$ " " !3$2, 0".$2y$ " 0 ! y$ " 0.

6 " $2x$ ! x$ "6$2

" 3$2.

3 "$22

x$ #$22

y$3 "$22

x$ !$22

y$

3 " x$ sin 45& # y$ cos 45&3 " x$ cos 45& ! y$ sin 45&

y " x$ sin % # y$ cos %x " x$ cos % ! y$ sin %

!3, 3"% " 45&;

73. x3 ! 16x " x!x2 ! 16" " x!x ! 4"!x # 4" 74. x2 # 14x # 49 " !x # 7"2

75.

" 2x!x ! 6"2

2x3 ! 24x2 # 72x " 2x!x2 ! 12x # 36" 76.

" x!3x # 2"!2x ! 5"

6x3 ! 11x2 ! 10x " x!6x2 ! 11x ! 10"

77.

" 2!2x # 3"!4x2 ! 6x # 9"

16x3 # 54 " 2!8x3 # 27" 78.

" !4 ! x"!x # i"!x ! i"

" !4 ! x"!x2 # 1"

4 ! x # 4x2 ! x3 " !4 ! x" # x2!4 ! x"

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3.

Hyperbola !y$ "2

2!

!x$ "2

2" 1,

%x$ ! y$

$2 &%x$ # y$

$2 & # 1 " 0

xy # 1 " 0

"x$ # y$

$2 " x$%$2

2 & # y$%$22 & y " x$ sin

'

4# y$ cos

'

4

"x$ ! y$

$2 " x$%$2

2 & ! y$%$22 & x " x$ cos

'

4! y$ sin

'

4

cot 2% "A ! C

B" 0 ! 2% "

'

2 ! % "

'

4

A " 0, B " 1, C " 0

!4 !3 !2 4

!4

!3

!2

4y " x "

x

yxy # 1 " 0

4.

, Hyperbola !x$ "2

4!

!y$ "2

4" 1

!x$ "2 ! !y$ "2

2" 2

%x$ ! y$

$2 &%x$ # y$

$2 & ! 2 " 0

xy ! 2 " 0

y " x$ sin '4

# y$ cos '4

" x$%$22 & # y$%$2

2 & "x$ # y$

$2

x " x$ cos '4

! y$ sin '4

" x$%$22 & ! y$%$2

2 & "x$ ! y$

$2

cot 2% "A ! C

B" 0 ! 2% "

'

2 ! % "

'

4 468

10

64 8 10

!8!10

x"y"

x

yxy ! 2 " 0, A " 0, B " 1, C " 0

5.

"$22

!x$ ! y$ "

" x$%$22 & ! y$%$2

2 & x " x$ cos

'4

! y$ sin '4

cot 2% "A ! C

B" 0 ! 2% "

'2

! % "'4

A " 1, B " !4, C " 1!4!6!8 4 6 8

!6

!8

4

6

8

x

y

y " x "

x2 ! 4xy # y2 # 1 " 0

y " x$ sin '4

# y$ cos '4

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5. —CONTINUED—

, Hyperbola !x$ "2 !!y$ "2

1#3" 1

!!x$ "2 # 3!y$ "2 " !1

12

!x$ "2 ! x$y$ #12

! y$ "2 ! 2'!x$ "2 ! !y$ "2( #12

!x$ "2 # x$y$ #12

!y$ "2 # 1 " 0

)$22

!x$ ! y$ "*2

! 4)$22

!x$ ! y$ "$22

!x$ # y$ "* # )$22

!x$ # y$ "*2

# 1 " 0

x2 ! 4xy # y2 # 1 " 0

6.

, Hyperbola %y$ #

3$22 &2

10!%x$ !

$22 &2

10" 1

%x$ !$22 &2

! %y$ #3$2

2 &2

" !10

)!x$ "2 ! $2x$ # %$22 &2* ! )!y$ "2 # 3$2y$ # %3$2

2 &2* " !6 # %$22 &2

! %3$22 &2

!x$ "2

2!

!y$ "2

2#

x$$2

!y$

$2!

2x$

$2!

2y$

$2# 3 " 0

%x$ ! y$

$2 &%x$ # y$

$2 & # %x$ ! y$

$2 & ! 2%x$ # y$

$2 & # 3 " 0

xy # x ! 2y # 3 " 0

"x$ ! y$

$2 "

x$ # y$

$2

" x$%$22 & ! y$%$2

2 & " x$%$22 & # y$%$2

2 & x " x$ cos

'

4! y$ sin

'

4 y " x$ sin

'

4# y$ cos

'

4

cot 2% "A ! C

B" 0 ! 2% "

'

2 ! % "

'

4

A " 0, B " 1, C " 0

x

x"y"

!4 4!6!8 6

4

6

8

!4

!6

!8

yxy # x ! 2y # 3 " 0

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7.

!x$ ! 3$2"2

16!

!y$ ! $2 "2

16" 1, Hyperbola

!x$ ! 3$2 "2! !y$ ! $2 "2

" 16

'!x$ "2 ! 6$2x$ # !3$2 "2( ! '!y$ "2 ! 2$2y$ # !$2 "2( " 0 # !3$2 "2! !$2 "2

!x$ "2

2!

!y$ "2

2! $2x$ ! $2y$ ! 2$2x$ # 2$2y$ " 0

%x$ ! y$

$2 &%x$ # y$

$2 & ! 2%x$ # y$

$2 & ! 4%x$ ! y$

$2 & " 0

xy ! 2y ! 4x " 0

"x$ ! y$

$2

" x$%$22 & ! y$%$2

2 & x " x$ cos

'

4! y$ sin

'

4

cot 2% "A ! C

B" 0 ! 2% "

'

2 ! % "

'

4

A " 0, B " 1, C " 0

x

x"

y "4

6

8

!4

!4 2 4 6 8

yxy ! 2y ! 4x " 0

"x$ # y$

$2

" x$%$22 & # y$%$2

2 & y " x$ sin

'

4# y$ cos

'

4

8.

—CONTINUED—

"x$ ! 3y$

$10 "

3x$ # y$

$10

" x$% 1$10& ! y$% 3

$10& " x$% 3$10& # y$% 1

$10& x " x$ cos % ! y$ sin % y " x$ sin % # y$ cos %

cos % "$1 # cos 2%

2"$1 # !!4#5"

2"

1$10

sin % "$1 ! cos 2%

2"$1 ! !!4#5"

2"

3$10

cos 2% " !45

cot 2% "( ! C

B" !

43

! % + 71.57&

A " 2, B " !3, C " !2

x

x"

y" 2

4

!4

!4 !2 4

y2x2 ! 3xy ! 2y2 # 10 " 0

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8. —CONTINUED—

, Hyperbola !x$ "2

4!

!y$ "2

4" 1

!52

!x$ "2 #52

!y$ "2 " !10

!x$ "2

5!

6x$y$

5#

9!y$ "2

5!

9!x$ "2

10#

24x$y$

10#

9!y$ "2

10!

9!x$ "2

5!

6x$y$

5!

!y$ "2

5# 10 " 0

2%x$ ! 3y$

$10 &2

! 3%x$ ! 3y$

$10 &%3x$ # y$

$10 & ! 2%3x$ # y$

$10 &2

# 10 " 0

2x2 ! 3xy ! 2y2 # 10 " 0

9.

!x$ "2

6#

!y$ "2

3#2" 1, Ellipse

2!x$ "2 # 8!y$ "2 " 12

52

!x$ "2 ! 5x$y$ #52

!y$ "2 ! 3!x$ "2 # 3!y$ "2 #52

!x$ "2 # 5x$y$ #52

!y$ "2 " 12

5)$22

!x$ ! y$ "*2

! 6)$22

!x$ ! y$ " $22

!x$ # y$ "* # 5)$22

!x$ # y$ "*2

" 12

5x2 ! 6xy # 5y2 ! 12 " 0

y " x$ sin '4

# y$ cos '4

"$22

!x$ # y$ "

x " x$ cos '4

! y$ sin '4

"$22

!x$ ! y$ "

% "'4

!2% "'2

!cot 2% "A ! C

B" 0

A " 5, B " !6, C " 5 x"y"

2

2

3

!3

!3!4

!4

4

3 4x

y5x2 ! 6xy # 5y2 ! 12 " 0

10.

—CONTINUED—

"$3x$ ! y$

2 "

x$ # $3y$

2

" x$%$32 & ! y$%1

2& " x$%12& # y$%$3

2 &x " x$ cos

'

6! y$ sin

'

6 y " x$ sin

'

6# y$ cos

'

6

cot 2% "A ! C

B"

1$3

! 2% "'

3 ! % "

'

6

A " 13, B " 6$3, C " 7

!3 !2 2 3

!3

!2

3

x

y "

x "

y13x2 # 6$3xy # 7y2 ! 16 " 0

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10. —CONTINUED—

, Ellipse !x$ "2

1#

!y$ "2

4" 1

16!x$ "2 # 4!y$ "2 " 16

!18!y$ "2

4#

7!x$ "2

4#

7$3x$y$

2#

21!y$ "2

4! 16 " 0

39!x$ "2

4!

13$3x$y$

2#

13!y$ "2

4#

18!x$ "2

4#

18$3x$y$

4!

6$3x$y$

4

13%$3x$ ! y$

2 &2

# 6$3%$3x$ ! y$

2 &%x$ # $3y$

2 & # 7%x$ # $3y$

2 &2

! 16 " 0

13x2 # 6$3xy # 7y2 ! 16 " 0

11.

Parabola x$ " !!y$ "2,

4(y$ )2# 4x$ " 0

# x$ ! $3y$ # 3x$ # $3y$ " 0

3(x$ ) 2

4!

6$3x$y$

4#

9(y$ ) 2

4!

6(x$ )2

4#

4$3x$y$

4#

6(y$ )2

4#

3(x$ ) 2

4#

2$3x$y$

4#

(y$ ) 2

4

# 2$3%$3x$ # y$

2 & " 0

3%x$ ! $3y$

2 &2

! 2$3%x$ ! $3y$

2 &%$3x$ # y$

2 & # %$3x$ # y$

2 &2

# 2%x$ ! $3y$

2 &

3x2 ! 2$3xy # y2 # 2x # 2$3y " 0

y " x$ sin % # y$ cos % "$3x$ # y$

2

" x$%12& ! y$%$3

2 & "x$ ! $3y$

2

x " x$ cos 60& ! y$ sin 60&

cot 2% "A ! C

B" !

1$3

! % " 60&

A " 3, B " !2$3, C " 1x

x"y"

2!4!6

2

!2

!4

y3x2 ! 2$3xy # y2 # 2x # 2$3y " 0

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12.

Parabola

!y$ "2 " 4!x$ ! 1"

25!y$ "2 ! 100x$ # 100 " 0

#81!y$ "2

25! 36x$ # 48y$ ! 64x$ ! 48y$ # 100 " 0

144!x$ "2

25!

384x$y$

25#

256!y$ "2

25!

288!x$ "2

25#

168x$y$

25#

288!y$ "2

25#

144!x$ "2

25#

216x$y$

25

16%3x$ ! 4y$

5 &2

! 24%3x$ ! 4y$

5 &%4x$ # 3y$

5 &# 9%4x$ # 3y$

5 &2

! 60%3x$ ! 4y$

5 &! 80%4x$ # 3y$

5 &# 100 " 0

16x2 ! 24xy # 9y2 ! 60x ! 80y # 100 " 0

y " x$ sin % # y$ cos % " x$%45& # y$%3

5& "4x$ # 3y$

5

x " x$ cos % ! y$ sin % " x$%35& ! y$%4

5& "3x$ ! 4y$

5

cos % "$1 # cos 2%

2"$1 # !!7#25"

2"

35

sin % "$1 ! cos 2%

2"$1 ! !!7#25"

2"

45

cos 2% " !725

cot 2% "( ! C

B" !

724

! % + 53.13&

A " 16, B " !24, C " 9

x1 2 3 4 5 6

1

x"

y"

y16x2 ! 24xy # 9y2 ! 60x ! 80y # 100 " 0

13.

—CONTINUED—

"3x$ ! 4y$

5

" x$%35& ! y$%4

5& x " x$ cos % ! y$ sin %

cos % "$1 # cos 2%

2"$1 # !!7#25"

2"

35

sin % "$1 ! cos 2%

2"$1 ! !!7#25"

2"

45

cos 2% " !725

cot 2% "A ! C

B" !

724

! % + 53.13&

A " 9, B " 24, C " 16

x

x "y"

!4 42

!2

2

4

6

y9x2 # 24xy # 16y2 # 90x ! 130y " 0

"4x$ # 3y$

5

" x$%45& # y$%3

5& y " x$ sin % # y$ cos %

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13. —CONTINUED—

Parabola !x$ ! 1"2 " 6%y$ #16&,

!x$ "2 ! 2x$ # 1 " 6y$ # 1

25!x$ "2 ! 50x$ ! 150y$ " 0

#144!y$"2

25# 54x$ ! 72y$ ! 104x$ ! 78y$ " 0

81!x$ "2

25!

216x$y$

25#

144!y$ "2

25#

288!x$ "2

25!

168x$y$

25!

288!y$ "2

25#

256!x$ "2

25#

384x$y$

25

! 130%4x$ # 3y$5 & " 0

9%3x$ ! 4y$

5 &2

# 24%3x$ ! 4y$

5 &%4x$ # 3y$

5 & # 16%4x$ # 3y$

5 &2

# 90%3x$ ! 4y$

5 &

9x2 # 24xy # 16y2 # 90x ! 130y " 0

14.

Parabola

!x$ "2 " 4y$,

25!x$ "2 ! 100y$ " 0

#144!y$ "2

25# 48x$ ! 64y$ ! 48x$ ! 36y$ " 0

81!x$ "2

25!

216x$y$

25#

144!y$ "2

25#

288!x$ "2

25!

168x$y$

25!

288!y$ "2

25#

256!x$ "2

25#

384x$y$

25

9%3x$ ! 4y$

5 &2

# 24%3x$ ! 4y$

5 &%4x$ # 3y$

5 & # 16%4x$ # 3y$

5 &2

# 80%3x$ ! 4y$

5 & ! 60%4x$ # 3y$

5 & " 0

9x2 # 24xy # 16y2 # 80x ! 60y " 0

y " x$ sin % # y$ cos % " x$%45& # y$%3

5& "4x$ # 3y$

5

x " x$ cos % ! y$ sin % " x$%35& ! y$%4

5& "3x$ ! 4y$

5

x " x$ cos % ! y$ sin % y " x$ sin % # y$ cos %

cos % "$1 # cos 2%

2"$1 # !!7#25"

2"

35

sin % "$1 ! cos 2%

2"$1 ! !!7#25"

2"

45

cos 2% " !725

cot 2% "( ! C

B" !

724

! % + 53.13&

A " 9, B " 24, C " 16

x

y

!1

1

2

3

x "

y "

!3 !2 !1 1

9x2 # 24xy # 16y2 # 80x ! 60y " 0

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15.

Solve for in terms of

y " !32

x ±$80 # 5x2

2

%y #32

x&2" 20 #

5x2

4"

80 # 5x2

4

y2 # 3xy #9x2

4" 20 ! x2 #

9x2

4

y2 # 3xy " 20 ! x2

x:y

cot 2% "A ! C

B"

1 ! 13

" 0 ! % "'4

" 45&

x2 # 3xy # y2 " 20

Graph and

!8

!12 12

8

y2 " !3x2

!$80 # 5x2

2.y1 " !

3x2

#$80 # 5x2

2

16.

To graph conic with a graphing calculator, we needto solve for in terms of

Graph and

!6

!9 9

6

y2 " x !$4 #x2

2.

y1 " x #$4 #x2

2

y " x ± $4 #x2

2

y ! x " ±$4 #x2

2

!y ! x"2 " 4 #x2

2

y2 ! 2xy # x2 " 4 !x2

2# x2

x2 ! 4xy # 2y2 " 8

x.y

% + 37.98&

2% + 75.96&

tan 2% " 4

1

tan 2%"

14

cot 2% "A ! C

B"

1 ! 2!4

"14

A " 1, B " !4, C " 2

x2 ! 4xy # 2y2 " 8 17.

Solve for in terms of by completing the square.

Graph and

!6

!9 9

6

y2 "16x ! 5$15x2 ! 21

7.

y1 "16x # 5$15x2 ! 21

7

y "16x ± 5$15x2 ! 21

7

y "167

x ± $375x2 ! 52549

%y !167

x&2

"375x2 ! 525

49

y2 !327

xy #25649

x2 "11949

x2 !52549

#25649

x2

y2 !327

xy "177

x2 !757

!7y2 # 32xy " !17x2 # 75

xy

cot 2% "A ! C

B"

17 # 732

"2432

"34

! % + 26.57&

17x2 # 32xy ! 7y2 " 75

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18.

Solve for in terms of by completing the square:

Graph and

y2 "!18x ! $1300 ! 676x2

25.

y1 "!18x # $1300 ! 676x2

25

y "!18x ± $1300 ! 676x2

25

y #1825

x " ±$1300 ! 676x2

625

%y #1825

x&2

"1300 ! 676x2

625

y2 #3625

xy #324625

x2 "5225

!4025

x2 #324625

x2

y2 #3625

xy "5225

!4025

x2

25y2 # 36xy " 52 ! 40x2

xy

% + 33.69&

2% + 67.38&

tan 2% "125

!2

!3 3

2 1

tan 2%"

512

cot 2% "A ! C

B"

40 ! 2536

"512

A " 40, B " 36, C " 25

40x2 # 36xy # 25y2 " 52 19.

Solve for in terms of

Graph and

!6

!9 9

6

y2 " !3x !$20x2 # 25

2.

y1 " !3x #$20x2 # 25

2

y " !3x ±$20x2 # 25

2

!y # 3x"2 " 5x2 #254

"20x2 # 25

4

y2 # 6xy # 9x2 " !4x2 #254

# 9x2

y2 # 6xy " !4x2 #254

8y2 # 48xy " !32x2 # 50

x:y

cot 2% "A ! C

B"

32 ! 848

"12

! % + 31.72&

32x2 # 48xy # 8y2 " 50

20.

—CONTINUED—

% + 33.69&

2% + 67.38&

tan 2% "125

1

tan 2%"

512

cot 2% "A ! C

B"

4 ! 9!12

"512

A " 4, B " !12, C " 9

4x2 ! 12xy # 9y2 # !4$13 ! 12"x ! !6$13 # 8"y " 91

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20. —CONTINUED—

Solve for in terms of with the Quadratic Formula:

Graph and

y2 "12x # 6$13 # 8 ! $624x # 3808 # 96$13

18.

y1 "12x # 6$13 # 8 # $624x # 3808 # 96$13

18!6

!9 27

18 "!12x # 6$13 # 8" ± $624x # 3808 # 96$13

18

y "!12x # 6$13 # 8" ± $!12x # 6$13 # 8"2 ! 4!9"!4x2 # 4$13x ! 12x ! 91"

18

y "!b ± $b2 ! 4ac

2a

a " 9, b " !!12x # 6$13 # 8", c " 4x2 # 4$13x ! 12x ! 91

9y2 ! !12x # 6$13 # 8"y # !4x2 # 4$13x ! 12x ! 91" " 0

4x2 ! 12xy # 9y2 # !4$13 ! 12"x ! !6$13 # 8"y " 91

xy

21.

The graph is a hyperbola.

Matches graph (e).

cot 2% "A ! C

B" 0 ! % " 45&

B2 ! 4AC " 1 !

xy # 4 " 0 22.

The graph is a line. Matches graph (b).

y " !x

x # y " 0

!x # y"2 " 0

x2 # 2xy # y2 " 0

23.

Matches graph (f).

cot 2% "A ! C

B" !

43

! % + !18.43&

" 25 ! The graph is a hyperbola.

B2 ! 4AC " !3"2 ! 4!!2"!2"

!2x2 # 3xy # 2y2 # 3 " 0 24.

The graph is an ellipse or circle.

Matches graph (a).

cot 2% "A ! C

B"

1 ! 3!1

" 2 ! % + 13.28&

B2 ! 4AC " !!1"2 ! 4!1"!3" " !11

A " 1, B " !1, C " 3

x2 ! xy # 3y2 ! 5 " 0

25.

Matches graph (d).

cot 2% "A ! C

B" 1 ! % " 22.5&

" !8 ! The graph is an ellipse or circle.

B2 ! 4AC " !2"2 ! 4!3"!1"

3x2 # 2xy # y2 ! 10 " 0 26.

The graph is a parabola.

Matches graph (c).

cot 2% "A ! C

B"

1 ! 4!4

"34

! % + 26.57&

B2 ! 4AC " !!4"2 ! 4!1"!4" " 0

A " 1, B " !4, C " 4

x2 ! 4xy # 4y2 # 10x ! 30 " 0

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27.

(a) Parabola

(b)

"24x # 40 ± $3000x # 1600

18

y "!24x # 40" ± $!24x # 40"2 ! 4!9"!16x2 ! 30x"

2!9"

9y2 ! !24x # 40"y # !16x2 ! 30x" " 0

B2 ! 4AC " !!24"2 ! 4!16"!9" " 0 !

16x2 ! 24xy # 9y2 ! 30x ! 40y " 0

(c)

!2

!4 8

6

28. (a)

(b)

(c)

!8

!12 12

8

"4x ± $24x2 ! 48

!4

y "4x ± $16x2 ! 4!!2"!x2 ! 6"

!4

!2y2 ! 4xy # x2 ! 6 " 0

> 0 ! Hyperbola

" 16 # 8 " 24

B2 ! 4AC " !!4"2 ! 4!1"!!2" 29.

(a)

(b)

(c)

!4

!6 6

4

"8x ±$1260 ! 356x2

14

y "8x ± $!!8x"2 ! 4!7"!15x2 ! 45"

14

7y2 ! 8xy # !15x2 ! 45" " 0

" !356 ! Ellipse or circle

B2 ! 4AC " !!8"2 ! 4!15"!7"

15x2 ! 8xy # 7y2 ! 45 " 0

30. (a)

(b)

(c)

!6

!10 8

6

"!4 ! 4x" ±$!24x2 ! 92x # 416

10

y "!4 ! 4x" ± $!4x ! 4"2 ! 4!5"!2x2 # 3x ! 20"

10

5y2 # !4x ! 4"y # !2x2 # 3x ! 20" " 0

" !24 < 0 ! Ellipse or circle

B2 ! 4AC " 42 ! 4!2"!5" 31.

(a)

(b)

(c)

!6

!10 8

6

"6x ±$56x2 # 80x ! 440

!10

y "6x ± $!!6x"2 ! 4!!5"!x2 # 4x ! 22"

!10

!5y2 ! 6xy # !x2 # 4x ! 22" " 0

" 56 ! Hyperbola

B2 ! 4AC " !!6"2 ! 4!1"!!5"

x2 ! 6xy ! 5y2 # 4x ! 22 " 0

32. (a)

(c)

!7

!10 2

1

" 0 ! Parabola

B2 ! 4AC " !!60"2 ! 4!36"!25" (b)

"60x ! 9 ±$!1080x # 81

50

y "!60x ! 9" ± $!9 ! 60x"2 ! 100!36x2"

50

25y2 # !9 ! 60x"y # 36x2 " 0

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33.

(a) Parabola

(b)

(c)

!5

!4 8

3

"1 ! 4x ±$72x # 49

8

y "!1 ! 4x" ± $!4x ! 1"2 ! 4!4"!x2 ! 5x ! 3"

8

4y2 # !4x ! 1"y # !x2 ! 5x ! 3" " 0

!B2 ! 4AC " 42 ! 4!1"!4" " 0

x2 # 4xy # 4y2 ! 5x ! y ! 3 " 0 34. (a)

(b)

(c)

!2

!3 3

2

"!x ! 1 ±$!15x2 ! 14x # 65

8

y "!!x # 1" ± $!x # 1"2 ! 4!4"!x2 # x ! 4"

8

4y2 # !x # 1"y # !x2 # x ! 4" " 0

" !15 < 0 ! Ellipse or circle

B2 ! 4AC " 1 ! 4!1"!4"

35.

Two intersecting lines

y " ±4x

y2 " 16x2

!1 1 2 3

3

!2!3x

y y2 ! 16x2 " 0 36.

Point at

!4 !3 !2 !1 1 2 3 4

!4

!3

!2

2

1

3

4

x

(1, !3)

y!1, !3"

!x ! 1"2 # !y # 3"2 " 0

!x2 ! 2x # 1" # !y2 # 6y # 9" " !10 # 1 # 9

x2 # y2 ! 2x # 6y # 10 " 0

37.

Two parallel lines

x

y

!1!2!3 1 2 3

1

2

3

!2

!3

y " !x ± 1

x # y " ±1

!x # y"2 " 1

!x # y"2 ! 1 " 0

x2 # 2xy # y2 ! 1 " 0 38.

Two lines

!2 !1 1 2 3 4

!2

1

2

3

4

x

y

y " !5 ± 2$6 "x y " 5x ± 2$6x

y ! 5x " ±$24x2

!y ! 5x"2 " 24x2

y2 ! 10xy # 25x2 " 25x2 ! x2

x2 ! 10xy # y2 " 0

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39.

Adding:

For

is impossible.

Solutions: !1, $3 ", !1, !$3 "x " !4

x " 1, y " ±$3.

x " 1, !4! !x # 4"!x ! 1" " 0

x2 # 3x ! 4 " 0

3x ! y2 " 0

x2 # y2 " 4 40.

No solution

"36 ± $!864

10

y "36 ± $362 ! 4!5"!108"

10

5y2 ! 36y # 108 " 0

4!27 ! y2" # 9y2 ! 36y " 0

x2 # y2 ! 27 " 0 ! x2 " 27 ! y2

4x2 # 9y2 ! 36y " 0

41.

x " !8

24x " !192

24x # 192 " 0

4x2 # y2 # 40x ! 24y # 208 " 0

!4x2 ! y2 ! 16x # 24y ! 16 " 0

43.

x " 0, 12! x!x ! 12" " 0

x2 ! 12x " 0

2x2 ! 24x " 0

x2 # y2 ! 12x ! 16y # 64 " 0

x2 ! y2 ! 12x # 16y ! 64 " 0

42.

When

Points of intersection: !6, !8", !14, !8"

y " !8

!y # 8"2 " 0

y2 # 16y # 64 " 0

!4y2 ! 64y ! 256 " 0

62 ! 4y2 ! 20!6" ! 64y ! 172 " 0

x " 6:

x " 6 or x " 14

!17x ! 238"!x ! 6" " 0

17x2 !340x # 1428 " 0

16x2 # 4y2 ! 320x # 64y # 1600 " 0 ! 16!x ! 10"2 # 4!y # 8"2 " 256

x2 ! 4y2 ! 20x ! 64y ! 172 " 0 ! !x ! 10"2 ! 4!y # 8"2 " 16

When

y " !8

!y # 8"2 " 0

y2 # 16y # 64 " 0

4y2 # 64y # 256 " 0

142 ! 4y2 ! 20!14" ! 64y ! 172 " 0

x " 14:

For

Solutions: !0, 8", !12, 8"

y " 8! !y2 # 16y ! 64 " 0

144 ! y2 ! 12!12" # 16y ! 64 " 0

x " 12:

For

Solution: !!8, 12"

! y " 12

!y ! 12"2 " 0

y2 ! 24y # 144 " 0

!y2 # 24y ! 144 " 0

!4!64" ! y2 ! 16!!8" # 24y ! 16 " 0

x " !8:

For

y " 8! !y ! 8"2 " 0

y2 ! 16y # 64 " 0

!y2 # 16y ! 64 " 0

x " 0:

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44.

When

Point of intersection: !1, 0"

x " 1

!x ! 1"2 " 0

x2 ! 2x # 1 " 0

x2 # 4!0"2 ! 2x ! 8!0" # 1 " 0

y " 0:

y " 0 or y " 3

4y!y ! 3" " 0

4y2 !12y " 0

!x2 # 2x ! 4y ! 1 " 0 ! y " ! 14!x ! 1"2

x2 # 4y2 ! 2x ! 8y # 1 " 0 ! !x ! 1"2 # 4!y ! 1"2 " 4

When

No real solution

x2 ! 2x # 13 " 0

!x2 # 2x ! 4!3" ! 1 " 0

y " 3:

45.

When

No real solution

The point of intersection is In standard form the equations are:

x2

25#

y2

16" 1

x2

4#

!y ! 12"2

64" 1

!0, 4".

16x2 " !225

16x2 # 25!!5"2 ! 400 " 0

y " !5:

y " !5 or y " 4

24!y # 5"!y ! 4" " 0

24y2 # 24y ! 480 " 0

16x2 # 25y2 ! 400 " 0

!16x2 ! y2 # 24y ! 80 " 0

When

x " 0

16x2 " 0

16x2 # 25!4"2 ! 400 " 0

y " 4:

46.

When

Points of intersection: !5, 8 # 4$21 ", !5, 8 ! 4$21 ", !!2, 8"

y " 8 ± 4$21

y2 ! 16y ! 272 " 0

y2 ! 48!5" ! 16y ! 32 " 0

x " 5:

x " 5 or x " !2

!x ! 5"!x # 2" " 0

16!x2 ! 3x ! 10" " 0

16x2 ! 48x ! 160 " 0

y2 ! 48x ! 16y ! 32 " 0 ! !y ! 8"2 ! 48x " 96

16x2 ! y2 # 16y ! 128 " 0 ! 16x2 ! !y ! 8"2 " 64

When

y " 8

!y ! 8"2 " 0

y2 ! 16y # 64 " 0

y2 ! 48!!2" ! 16y ! 32 " 0

x " !2:

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47.

Two solutions: !$3, !2$3 ", !!$3, 2$3 " x2 " 3 ! x " ±$3

!2x2 # 6 " 0

2x2 ! !!2x"2 # 6 " 0

2x # y " 0 ! y " !2x

2x2 ! y2 # 6 " 0 48.

Solutions: !0, 2", !43, 23"

x " 43 ! y " 2 ! 4

3 " 23

x " 0 ! y " 2

3x!3x ! 4" " 0

9x2 ! 12x " 0

6x2 # 3!4 ! 4x # x2" ! 12 " 0

6x2 # 3!2 ! x"2 ! 12 " 0

x # y ! 2 " 0 ! y " 2 ! x

6x2 # 3y2 ! 12 " 0

49.

impossible

One solution: !8, 0"

x " 7 ! y2 " !2

x " 8 ! y2 " 0 ! !8, 0"

!x ! 8"!x ! 7" " 0

x2 ! 15x # 56 " 0

10x2 ! 150x # 560 " 0

10x2 ! 25!2x ! 16" ! 100x # 160 " 0

y2 ! 2x # 16 " 0 ! y2 " 2x ! 16

10x2 ! 25y2 ! 100x # 160 " 0 50.

From Equation 1:

In Equation 2:

Solution: !2, 3"

x " 45 ! y2 ! 6y " !321

25 No solution

! !y ! 3"2 " 0 ! y " 3

x " 2 ! y2 ! 6y " !9

!x ! 2"!5x ! 4" " 0

5x2 ! 14x # 8 " 0

!10x2 # 28x ! 16 " 0

2x2 # 4x ! 43 ! !12x2 ! 24x ! 27" " 0

2x2 # 4x ! 43 ! !3y2 ! 18y" " 0

3y2 ! 18y " 12x2 ! 24x ! 27

y2 ! 6y " 4x2 ! 8x ! 9

2x2 ! 3y2 # 4x # 18y ! 43 " 0

4x2 ! y2 ! 8x # 6y ! 9 " 0

51.

x " 0 or x " !3

x(x # 3)!x2 ! 7x # 20" " 0

x4 ! 4x3 ! x2 # 60x " 0

x4 ! 4x3 # 4x2 # 4x2 # 24x # 36 " 9x2 ! 36x # 36

x2!x2 ! 4x # 4" # 4!x2 # 6x # 9" " 9!x2 ! 4x # 4"

x2!x ! 2"2 # 4!!x ! 3"2 " 9!x ! 2"2

x2 # 4%!x ! 3x ! 2 &

2

" 9

x2 # 4y2 ! 9 " 0

xy # x ! 2y # 3 " 0 ! y "!x ! 3x ! 2

Note: has no real solution.

When

When

The points of intersection are !0, 32", !!3, 0".

y "!!!3" ! 3

!3 ! 2" 0x " !3:

y "!0 ! 30 ! 2

"32

x " 0:

x2 ! 7x # 20 " 0

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52.

When :

When :

Points of intersection: %16

!3 ! $30 ", 16

!3 # $30 "&%16

!3 # $30 ", 16

!3 ! $30 "&,

y " 1 !3 !$30

6"

3 #$306

x "3 !$30

6

y " 1 !3 #$30

6"

3 !$306

x "3 #$30

6

x "3 ±$30

6

12x2 ! 12x ! 7 " 0

5x2 ! 2x # 2x2 # 5 ! 10x # 5x2 ! 12 " 0

5x2 ! 2x # 2x2 # 5!1 ! 2x # x2" ! 12 " 0

5x2 ! 2x!1 ! x" # 5!1 ! x"2 ! 12 " 0

x # y ! 1 " 0 ! y " 1 ! x

5x2 ! 2xy # 5y2 ! 12 " 0

53. True.

If then B2 ! 4AC > 0.k < 14,

B2 ! 4AC " 1 ! 4k 54. False. See Example 2. However, A # C " A$ # C$.

55.

Asymptotes:

Intercepts: !0, 1"

x " 2, y " 0!2 !1 1 3

!4

!3

!2

2

3

4

x

(0, 1)

y

g!x" "2

2 ! x56.

Intercept:

Asymptotes:x " 2, y " !2

4 6 8

!6

!4

2

4

6

x(0, 0)

y!0, 0"

f !x" "2x

2 ! x" !2 #

42 ! x

57.

Slant asymptote:

Vertical asymptote:

Intercept: !0, 0"

!10 !5 5 10 15

!15

!10

!5

5

10

(0, 0)t

yt " 2

y " !t ! 2

h!t" "t2

2 ! t" !t ! 2 #

42 ! t

58.

Intercept:

Asymptotes:s " ±2, y " 0

%0, 12&

!4 !1 1 4

!4

!3

!2

1

2

3

4

s( )0 , 1

2

yg!s" "2

4 ! s2

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59. (a)

(b)

(c) A2 " )12

!35*)

12

!35* " )!5

12!18

19*

BA " )05

6!1*)

12

!35* " )12

330

!20*

AB " )12

!35*)

05

6!1* " )!15

2597* 60. (a)

(b)

(c) A2 " )10

5!2*)

10

5!2* " )1

0!5

4*

BA " ) 3!3

28*)

10

5!2* " ) 3

!311

!31*

AB " )10

5!2*)

3!3

28* " )!12

642

!16*

61. (a)

(b)

(c) does not exist.A2

BA " ) 3!4

5*'4 !2 5( " ) 12!16

20

!68

!10

15!20

25*AB " '4 !2 5() 3

!45* " '12 # 8 # 25( " '45(

62. (a)

(b)

(c) A2 " )!2164

!219

!2

!105

20*BA " )!9

29

!149

!1

02515*

AB " ) 827

!13

!102020

26

!13* 63.

x2

4

6

8

10

!8 !6 !4 !2!2

4

y

f !x" " ,x # 3, 64.

x2

2

4

8

10

!2!2

4 6 8 10

y

f !x" " ,x ! 4, # 1

65.

!1!2!3 1 2 3

1

3

!2

!1

!3

x

y

g!x" " $4 ! x2 66.

x2

2

4

6

8

10

!2!2

4 6 8 10

y

g!x" " $3x ! 2 67.

!2!4!6!8!10 2 6 8 10

2468

10121416

!4

x

y

h!t" " !!t ! 2"3 # 3

68.

x

y

!1!2!3!4!6 1 2

!4

!3

!2

!1

1

2

3

4

h!t" " 12!t # 4"3 69.

!1!2!3 1 2 5 6 7

123

!2!3!4

!7

x

y

f !t" " -t ! 5. # 1 70.

!1!2!3!4 1

1

2

3

4

2 3 4

!3

!2

x

y

f !t" " !2-t. # 3

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Section 9.5 Parametric Equations 825

71.

! 45.11

! 12"8#"12# sin 110"

Area ! 12ab sin C 72.

! 187.94

! 12"25#"16# sin 70"

Area ! 12ac sin B

73.

! 48.60

! $392 "17

2 #"32#"19

2 #Area ! $s"s # a#"s # b#"s # c#

s ! 12"11 $ 18 $ 10# ! 39

2 74.

! 310.39

! $852 "39

2 #"152 #"31

2 #Area ! $s"s # a#"s # b#"s # c#

s ! 12"23 $ 35 $ 27# ! 85

2

Section 9.5 Parametric Equations

! If f and g are continuous functions of t on an interval I, then the set of ordered pairs is a planecurve C. The equations and are parametric equations for C and t is the parameter.

! You should be able to graph plane curves with your graphing utility.

! To eliminate the parameter:

Solve for t in one equation and substitute into the second equation.

! You should be able to find the parametric equations for a graph.

y ! g"t#x ! f"t#"f"t#, g"t##

Vocabulary Check

1. plane curve, parametric equations, parameter 2. orientation

3. eliminating, parameter

1.

line

Matches (c).

y ! x $ 2,

y ! t $ 2

x ! t 2.

Parabola opening to the rightMatches (d).

x ! "y $ 2#2

y ! t # 2 ! t ! y $ 2

x ! t2 3.

parabola,

Matches (b).

x " 0y ! x2,

y ! t

x ! $t

4.

Matches (a).

y ! t $ 2 ! y !1x

$ 2

x !1t ! t !

1x

5.

Matches (f).

y !12

ex # 2

y !12

t # 2

x ! ln t # t ! ex 6.

Exponential curve on

Matches (e).

x $ 0

y ! ex2%4

y ! et

x ! #2$t ! t ! & x#2'

2

!x2

4

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11.

!1

!2

!1

!2 1 2

2

x

y

y ! #4x

x ! t, y ! #4t 12.

or

2

4

2

4

6

6!6

!6

!4

!4!2

x

y

x # 2y ! 0y !12

x

x ! t, y !12

t

7.

(a)

(b) Graph by hand.

Note: x " 0

!1

!2

!1

!2 1 2

1

2

x

y

x ! $t, y ! 2 # t

(c)

(d)

Parabola

In part (c), x " 0.

!1

!2

!1

!2 1 2

1

x

yy ! 2 # t ! 2 # x2,

!3

3

5!4

0 1 2 3 4

0 1 2

2 1 0 #2#1y

$3$2x

t

8.

(a)

(b)

!1!2!3!4!5 1 2 3 5

12345

!2!3!4!5

x

y

x ! 4 cos2 %, y ! 4 sin %

(c)

(d)

parabola

The graph is an entireparabola rather thanjust the right portion.

!1!2!3!4!5 1 2 3 5

123

5

!2!3

!5

x

y x4

$ y2

16! 1,

4x $ y2 ! 16 cos2 % $ 16 sin2 % ! 16

10!8

!6

6

0

0 2 4 2 0

0 42$2#2$2#4y

x

&2

&4

#&4

#&2

%

9. The graph opens upward, contains and is oriented left to right. Matches (b).

"1, 0#, 10. The orientation of the graph is clockwise and thecenter is Matches (c)."2, 3#.

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13.

!1!2!3!4!6 1 2!1

!2

!3

1

2

4

5

x

y

y !23

x $ 3

y ! 2&x $ 33 ' $ 1

t !x $ 3

3

x ! 3t # 3, y ! 2t $ 1 14.

2

4

2

4

6

6 8

!4

!4!2

!2x

y

3x $ 2y # 13 ! 0

y ! 2 $ 3&3 # x2 '

x ! 3 # 2t, y ! 2 $ 3t 15.

1 2

5

3!3 !2 !1!1

x

y

y ! 16x2

y ! "4x#2

x !14

t, y ! t2

16.

2 4

2

4

6

6!6

!6

!4 !2x

y

y ! x3

x ! t, y ! t3 17.

!1!2 1 2 3 4 5 6!1

1

2

4

3

x

y

y ! "x # 2#2

t ! x # 2

x ! t $ 2, y ! t2 18.

2 4

2

6 8 10!2!2

!4

!6

!8

!10

x

y

y ! 1 # x2, x " 0

y ! 1 # t

x ! $t

20.

Eliminating the parameter and

2

6

4

2

8

10

6 8 10!2!2

y

x

! (y # 3(. ! (" y # 2# # 1(

x ! (t # 1(t ! y # 2t,

y ! t $ 2

x ! (t # 1(19.

2 4

6

4

8

10

6 8 10!2!2

x

y

!12(x # 4(

! (x2 # 2( t !

x2

! y ! (t # 2(

x ! 2t, y ! (t # 2(

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21.

ellipsex2

4$

y2

9! 1,

x2

4$

y2

9! cos2 % $ sin2 % ! 1

&x2'

2! cos2 %, &y

3'2

! sin2 %

!1!3!4 1

1

2

4

3 4

!2

!4

!1

x

y

x ! 2 cos %, y ! 3 sin % 22.

x2 $y2

16! 1, ellipse

x2 $ &y4'

2! cos2 % $ sin2 % ! 1

!2!3!4!5 2 3 4 5

45

!4!5

x

y

x ! cos %, y ! 4 sin %

23.

!1!2 321 4 5 6 7 8

123456789

10

x

y

y !1x3 , x > 0, y > 0

y ! &1x'

3

y ! e3t ! y ! "et#3

x ! e#t ! 1x

! et 24.

1

1 2 3 4 5

2

3

4

5

x

y

y2 ! x, y > 0; y ! $x, x > 0

y ! et ! y2 ! e2t

x ! e2t

25.

!1

!3!2

!4!5

!2 32 4 5 6 7 8

12345

x

y

y ! ln x

y ! ln"x1%3#3

y ! 3 ln t ! y ! ln t3

x ! t3 ! x1%3 ! t 26.

1

1!1!2 2!3 3

2

3

4

5

6

x

y

y ! 2t2 ! 2"12 e x#2 ! 1

2 e2x

x ! ln 2t ! ex ! 2t ! t ! 12ex

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27.

8!1

!5

1

x ! 4 $ 3 cos %, y ! #2 $ sin % 28.

!6

10!2

2

x ! 4 $ 3 cos %, y ! #2 $ 2 sin %

29.

12!12

!8

8

x ! 4 sec %, y ! 2 tan % 30.

y ! tan %

!4

4

6!6

x ! sec %

31.

y ! ln"t2 $ 1#

!8

8

12!12

x !t2

32.

y ! 0.4t2

!5

35

20!40

x ! 10 # 0.01et

33. By eliminating the parameters in (a)–(d), we get . They differ from each other in restricted domain and in orientation.

(a) Domain:

Orientation: Left to right

(c) Domain:

Orientation: Right to left

0 < x < '

#' < x < '

y ! 2x $ 1

(b) Domain:

Orientation: Depends on

(d) Domain:


0 < x < '

%

#1 $ x $ 1

34. Each curve represents a portion of the line

(a)


(c)


y ! 3 # t ! 3 # &x # 22 ' ! 4 #

x2

x ! 2"t $ 1#, #' < x < '

y ! 4 # $t ! 4 #x2

, y $ 4

x ! 2$t, x " 0

2y $ x # 8 ! 0.

(b)


(d)

Orientation: Left to right for

Right to left for t > 0

t $ 0

y ! 4 $ t2 ! 4 #x2

x ! #2t2, x $ 0

y ! 4 # 3$t ! 4 #x2

x ! 2 3$t, #' < x < '

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49.

!4

4

6!6

x ! 2 cot %, y ! 2 sin2 % 50.

!4

4

6!6

y !3t2

1 $ t3x !3t

1 $ t3,

35.

! y # y1 ! &y2 # y1

x2 # x1'"x # x1#

y ! y1 $ & x # x1

x2 # x1'"y2 # y1#

t !"x # x1#"x2 # x1#

37.

"x # h#2

a2 $"y # k#2

b2 ! 1

x # ha

! cos %, y # k

b! sin %

y ! k $ b sin %

x ! h $ a cos %

36.

"x # h#2 $ " y # k#2 ! r2

cos2 % $ sin2 % ! "x # h#2

r2 $" y # k#2

r2 ! 1

y # kr

! sin %"x # h#

r! cos %,

y ! k $ r sin %

x ! h $ r cos %

38.

sec2 % # tan2 % !"x # h#2

a2 #"y # k#2

b2 ! 1

y # kb

! tan %x # h

a! sec %,

y ! k $ b tan %

x ! h $ a sec %

39.

y ! y1 $ t"y2 # y1# ! 4 $ t"#3 # 4# ! 4 # 7t

x ! x1 $ t"x2 # x1# ! 1 $ t"6 # 1# ! 1 $ 5t 40.

y ! k $ r sin % ! 5 $ 4 sin %

x ! h $ r cos % ! 2 $ 4 cos %

41. and

The center is so and

so and

This solution is not unique.y ! 3 sin %.

x ! 5 cos %cos2 % $ sin2 % ! 1 !x2

52 $y2

32,

k ! 0.h ! 0"0, 0#,

b ! $a2 # c2 ! 3.a ! 5, c ! 4, 42. and

The center is so and

so and

x ! $3 tan %.

y ! sec %sec2 % # tan2 % ! 1 !y2

1#

x2

3,

k ! 0.h ! 0"0, 0#,

b ! $c2 # a2 ! $3.a ! 1, c ! 2,

43.

Answers will vary.

x !15

t, y ! t # 3

x ! t, y ! 5t # 3

y ! 5x # 3 44.

Answers will vary.

x ! 2t, y ! 4 # 14t

x ! t, y ! 4 # 7t

y ! 4 # 7x 45.

Sample answers:

x ! t3, y !1t3

x ! t, y !1t

y !1x

48.

Sample answers:

x !12

t, y !t3

8$ t

x ! t, y ! t3 $ 2t

y ! x3 $ 2x46.

Sample answers:

x ! 2t, y !14t

x ! t, y !12t

y !1

2x47.

Sample answers:

x ! 2t, y ! 24t2 # 5

x ! t, y ! 6t2 # 5

y ! 6x2 # 5©

Hou

ghto

n M

ifflin

Com

pany

. All

right

s re

serv

ed.


51. Matches (b). 52. Matches (c). 53. Matches (d). 54. Matches (a).

55.

(a)

(b)

It is not a home run because when x ! 400.y < 10

0

30

4500

! 3 $ 38.0t # 16t2

y ! 3 $ "146.67 sin 15"#t # 16t2

x ! "146.67 cos 15"#t ! 141.7t

% ! 15"

y ! 3 $ "146.67 sin %#t # 16t2

x ! "146.67 cos %#t

! 146.67 ft%sec

100 miles%hour !100 mi(hr ) 5280 ft(mi

3600 sec(hr

x ! "v0 cos %#t, y ! h $ "v0 sin %#t # 16t2

56. (a)

(c) The horizontal distance is approximately 342.25 feet.

(d) You could use the Quadratic Formula to findthe zeros of The larger zero, 4.255, gives feet.x ! 342.25

y ! #16t2 $ "105 sin 40"#t $ 2.5.

! 2.5 $ "105 sin 40"#t # 16t2

y ! h $ "v0 sin %#t # 16t2

x ! "v0 cos %#t ! "105 cos 40"#t

(c)

Yes, it is a home run because when

(d) is the minimum angle.% ! 19.4"

x ! 400.y > 10

0

60

5000

! 3 $ 57.3t # 16t2

y ! 3 $ "146.67 sin 23"#t # 16t2

x ! "146.67 cos 23"#t ! 135.0t

% ! 23"

(b)

The maximum height is approximately 73.68 feet, when seconds.t ! 2.109

400

80

00

57. (a)

y ! 7 $ "sin 35"#v0t # 16t2

x ! "cos 35"#v0 t (b) If the ball is caught at time then:

! v0 !90

t1 cos 35"! 54.09 ft(sec

! t1 ! 2.03 seconds

! 16t12 ! 90 tan 35" $ 3

v0t1 !90

cos 35" ! #3 ! "sin 35"# 90

cos 35"# 16t12

4 ! 7 $ "sin 35"#v0t1 # 16t12.

90 ! "cos 35"#v0t1

t1,

(c)

Maximum height

(d) From part (b), t1 ! 2.03 seconds.

! 22 feet

0

24

900

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58. (a)

(b)

The maximum height is approximately 66.25 feet when seconds.

(c) The horizontal distance is approximately 222.35 feet.

(d) You could solve the equation for Then, feet.x ! 222.35

t ! 4.0696.y ! "85 sin 50"#t # 16t2 ! 0

t ! 2.035

240

80

00

y ! h $ "v0 sin %#t # 16t2 ! "85 sin 50"#t # 16t2

x ! "v0 cos %#t ! "85 cos 50"#t

59. True

first set

second set

y ! 9t2 $ 1 ! "3t#2 $ 1 ! x2 $ 1

x ! 3t

y ! t2 $ 1 ! x2 $ 1

x ! t

60. False. The graph of represents theportion of the line in the first quadrant.y ! x

x ! t2, y ! t2

61. False. For example, and does not represent as a function of x.y

y ! tx ! t2 62. False. The equations represent a line.

63. Sample answer:

y ! #2 sin %

x ! cos % 64. The graph is the same, but the orientation isreversed.

65.

Symmetric about the -axis

Even function

y

f"#x# !4"#x#2

"#x#2 $ 1!

4x2

x2 $ 1! f"x#

68.

No symmetry

Neither even nor odd

y ! x2 # 4x

"x # 2#2 ! y $ 4

66.

No symmetry


f "x# ! $x, x " 0

67.

No symmetry


y ! ex * e#x; e#x * #ex

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Section 9.6 Polar Coordinates

Section 9.6 Polar Coordinates 833

! In polar coordinates you do not have unique representation of points. The point can be represented by or by where n is any integer. The pole is represented by where

is any angle.

! To convert from polar coordinates to rectangular coordinates, use the following relationships.

! To convert from rectangular coordinates to polar coordinates, use the following relationships.

If is in the same quadrant as the point , then r is positive. If is in the opposite quadrant as the pointthen r is negative.

! You should be able to convert rectangular equations to polar form and vice versa.

!x, y", !!x, y"!

tan ! " y#x

r " ±$x2 # y2

y " r sin !

x " r cos !

!!0, !"!$r, ! ± !2n # 1"%"!r, ! ± 2n%"

!r, !"

1. Polar coordinates:

Rectangular coordinates: !0, 4"

y " 4 sin%%

2& " 4

x " 4 cos%%

2& " 0

%4, %

2& 2. Polar coordinates:

Rectangular coordinates: !0, $4"

x " 4 cos%3%

2 & " 0, y " 4 sin%3%

2 & " $4

%4, 3%

2 &


Rectangular coordinates: %$22

, $22 &

y " $1 sin%5%

4 & "$22

x " $1 cos%5%

4 & "$22

%$1, 5%

4 & 4. Polar coordinates:

Rectangular coordinates: !$2, $$2 "

y " 2 sin%$%4& " 2%$

$22 & " $$2

x " 2 cos%$%4& " 2%$2

2 & " $2

%2, $%4&

Vocabulary Check1. pole 2. directed distance, directed angle 3. polar

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5.

Three additional representations:

%$3, 5%6

$ %& " %$3, $%6&

%$3, 5%6

# %& " %$3, 11%

6 &%3,

5%6

$ 2%& " %3, $7%6 &

01 2 3

!2

3, 6!5( (

6.

Three additional points:

%2, $5%

4 &, %$2, 7%4 &, %$2,

$%4 &

01 2 3

!2

2,43!( (

7.


%1, $%3

$ %& " %1, $4%3 &

%1, $%3

# %& " %1, 2%3 &

%$1, $%3

# 2%& " %$1, 5%3 &

01 2 3

"1, "3!( (

!2 8.


%3, $%6&%$3,

5%6 &, %3,

11%6 &,

01 2 3 4

"3, " )) 76!

2!

9.


%$$3, 11%

6 &%$$3, $%6&,%$3, $

7%6 &,

0

3, )) 56!

1 2 3

2! 10.


%$5$2, 7%6 &%$5$2, $

5%6 &,%5$2,

%6&,

0

( )"2,5

1 3 5 7 9

116!2

!

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11.


%32

, %2&, %$

32

, 3%2 &, %$

32

, $%2&

32

0

, "

1 2 3

)) 32!2

! 12.

is the origin.


Any angle will do since r " 0."!

%0, 3%4 &, %0,

$5%4 &, %0,

7%4 &

%0, $%

4 &

01 2 3

!2

0,4!( ("


Rectangular coordinates:

2 4 60

2!

!2, $2$3"

y " 4 sin%$%

3& " $2$3

x " 4 cos%$%

3& " 2

%4, $%

3& 14. Polar coordinates:


1 2 3

2!

0

!$$3, $1"

y " 2 sin 7%6

" 2%$12& " $1

x " 2 cos 7%6

" 2%$$32 & " $$3

%2, 7%6 &



1 2 30

2!

%$22

, $22 &

y " $1 sin%$3%

4 & "$22

x " $1 cos%$3%

4 & "$22

%$1, $3%

4 & 16. Polar coordinates:


01 2 3

2!

%32

, 3$3

2 &

y " $3 sin%$2%3 & " $3%$

$32 & "

3$32

x " $3 cos%$2%3 & " $3%$

12& "

32

%$3, $2%3 & " %3,

%3&

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17. Polar coordinates: (origin!)


1 2 30

2!

!0, 0"

y " 0 sin%$7%

6 & " 0

x " 0 cos%$7%

6 & " 0

%0, $7%

6 & 18. Polar coordinates: (origin!)


01 2 3

2!

!0, 0"

y " 0 sin 5%4

" 0

x " 0 cos 5%4

" 0

%0, 5%4 &



1 2 30

2!

!$0.0024, 3"

y " $3 sin!$1.57" ' 3.000

x " $3 cos!$1.57" ' $0.0024

!$3, $1.57"19. Polar coordinates:


1 2 30

2!

!$1.004, 0.996"

y " $2 sin!2.36" ' 0.996

x " $2 cos!2.36" ' $1.004

!$2, 2.36"

21. !r, !" " %2, 2%

9 & ! !x, y" " !1.53, 1.29" 22. !r, !" " %4, 11%

9 & ! !x, y" " !$3.06, $2.57"

23. !r, !" " !$4.5, 1.3" ! !x, y" " !$1.204, $4.336" 24. !r, !" " !8.25, 3.5" ! !x, y" " !$7.726, $2.894"

25. !r, !" " !2.5, 1.58" ! !x, y" " !$0.02, 2.50" 26. !r, !" " !5.4, 2.85" ! !x, y" " !$5.17, 1.55"

27. !r, !" " !$4.1, $0.5" ! !x, y" " !$3.60, 1.97" 28. !r, !" " !8.2, $3.2" ! !x, y" " !$8.19, 0.48"

29. Rectangular coordinates:

Polar coordinates: !7, %", !$7, 0"

r " 7, tan ! " 0, ! " 0

"1"2"4"6"8 "3"5"7"9

1

1

"2"3"4"5

2

3

45

x

y!$7, 0"

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Polar coordinates:

1 2 3"3

"3

" 4

"5

"6

"2

"2"1

"1x

y

%5, 3%

2 &, %$5, %

2&

r " 5, tan ! undefined, ! "%

2

!0, $5" 31. Rectangular coordinates:

Polar coordinates:

1

2

1

2

3

3"3

"3

"2

"2"1

"1x

y

%$2, %

4&, %$$2, 5%

4 &

r " $2, tan ! " 1, ! "%

4

!1, 1"


Polar coordinates:

2

1

1 2"3"4

"4

"3

"2

"2"1

"1x

y

%3$2, 5%

4 &, %$3$2, %

4&

r " 3$2, tan ! " 1, ! "%

4

!$3, $3"


Polar coordinates:

x

y

"1"2 1 2 3 4 5 6"1

"2

"3

"4

"5

"6

1

2

%2, 11%

6 &, %$2, 5%

6 &

! "11%

6tan ! "

$1$3

,

r " $3 # 1 " 2

!$3, $1"


Polar coordinates:

1

2

1

2

3

3"3

"3

"2

"2"1

"1x

y

%$6, 5%

4 &, %$$6, %

4&

r " $3 # 3 " $6, tan ! " 1, ! "%

4

!$$3, $$3"

35.

Polar coordinates:

"3 3 6 9 12

"3

3

6

9

12

x

y

!10.8, 0.983", !$10.8, 4.124"

tan ! "96

"32

! ! ' 0.983

r " $62 # 92 " $117 ' 10.8

!x, y" " !6, 9"

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Polar coordinates:

4

2

"2"4"6

6

8

10

12

4 6 82"2

x

y

!$13, 4.318" !13, 1.176",

! ' 1.176tan ! " 125 ,r " $25 # 144 " 13,

!5, 12" 37.

!r, !" ' !$13, $0.588" ! " arctan!$2

3" ' $0.588

!x, y" " !3, $2" ! r " $32 # !$2"2 " $13

38. !x, y" " !$5, 2" ! !r, !" " !5.39, 2.76"

42. !x, y" " %74

, 32& ! !r, !" " !2.30, 0.71"

39.

!r, !" ' !$7, 0.857"

! " arctan% 2$3& ' 0.857

!x, y" " !$3, 2" ! r " $3 # 22 " $7

40. !x, y" " !3$2, 3$2" ! !r, !" " %6, %

4& ' !6.0, 0.785"

41.

!r, !" ' %176

, 0.490&

! " arctan%4#35#2& ' 0.490

!x, y" " %52

, 43& ! r "$%5

2&2

# %43&

2

"176

43.

r " 3

r2 " 9

x2 # y2 " 9 44.

r " 4

r2 " 16

x2 # y2 " 16 45.

r " 4 csc !

r sin ! " 4

y " 4

47.

r " 8 sec !

r cos ! " 8

x " 846.

! "%4

tan ! " 1

sin ! " cos !

r sin ! " r cos !

y " x 48.

r " a sec !

r cos ! " a

x " a

49.

r "2

6 sin ! $ 3 cos !

r!3 cos ! $ 6 sin !" " $2

3r cos ! $ 6r sin ! " $2

3x $ 6y # 2 " 0 50.

r "2

4 cos ! # 7 sin !

r!4 cos ! # 7 sin !" " 2

4r cos ! # 7r sin ! $ 2 " 0

4x # 7y $ 2 " 0

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51.

r2 " 8 csc 2!

r2 sin 2! " 8

r2!2 cos ! sin !" " 8

r2 cos ! sin ! " 4

!r cos !"!r sin !" " 4

xy " 4 52.

r2 " 12 sec ! csc ! " csc!2!"

2r2 " sec ! csc !

2r cos ! & r sin ! " 1

2xy " 1

53.

r2 " 9 cos!2!"

r2 " 9!cos2 ! $ sin2 !"

!r2"2 " 9!r2 cos2 ! $ r2 sin2 !"

!x2 # y2"2 " 9!x2 $ y2" 54.

or r "$4

1 # cos !r "

41 $ cos !

r " ±!r cos ! # 4"

!r cos ! # 4"2 " r2

r2 cos2 ! # 8r cos ! # 16 " r2

r2!1 $ cos2 !" $ 8r cos ! $ 16 " 0

r2 sin2 ! $ 8r cos ! $ 16 " 0

y2 $ 8x $ 16 " 0

55.

r " 6 cos !

r2 " 6r cos !

r2 $ 6r cos ! " 0

x2 # y2 $ 6x " 0 56.

r " 8 sin !

r!r $ 8 sin !" " 0

r2 $ 8r sin ! " 0

x2 # y2 $ 8y " 0 57.

r " 2a cos !

r!r $ 2a cos !" " 0

r2 $ 2ar cos ! " 0

x2 # y2 $ 2ax " 0

58.

r " 2a sin !

r!r $ 2a sin !" " 0

r2 $ 2a r sin ! " 0

x2 # y2 $ 2ay " 0 59.

" tan2 ! sec !

r "sin2 !cos3 !

sin2 ! " r cos3 !

!r sin !"2 " !r cos !"3

y2 " x3 60.

r "cos2 !sin3 !

" cot2 ! csc !

r2 cos2 ! " r3 sin3 !

x2 " y3

61.

x2 # y2 $ 6y " 0

x2 # y2 " 6y

r2 " 6r sin !

r " 6 sin ! 62.

x2 # y2 " 2x

r2 " 2r cos !

r " 2 cos ! 63.

y " $3x

$3 "yx

tan ! " tan 4%3

"yx

! "4%3

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64.

y # $3x " 0

yx

" $$3

tan ! " tan 5%3

" $$3

! "5%3

65.

y "$$3

3x

$$3

3"

yx

tan ! " tan 5%6

"yx

! "5%6

66.

y "$$3

3x

$$3

3"

yx

tan ! " tan 11%

6"

yx

! "11%

6

67. vertical line

x " 0

! "%2

, 68. horizontal line

y " 0

! " %, 69.

x2 # y2 " 16

r2 " 16

r " 4

70.

x2 # y2 " 100

r2 " 100

r " 10 71.

y " $3

r sin ! " $3

r " $3 csc ! 72.

x " 2

r cos ! " 2

r " 2 sec !

73.

!x2 # y2"3 " x2

x2 # y2 " x2#3

!x2 # y2"3#2 " x

r3 " r cos !

r2 " cos ! 74.

!x2 # y2)2 " 2xy

r4 " 2xy

r2 " 2 %yr&%

xr& "

2xyr2

r2 " sin 2! " 2 sin ! cos !

75.

!x2 # y2"2 " 6x2y $ 2y3

!x2 # y2"2 " 6!x2 # y2"y $ 8y3

r4 " 6r3 sin ! $ 8r3 sin3 !

r " 2!3 sin ! $ 4 sin3 !"

r " 2 sin 3! 76.

!x2 # y2"3#2 " 3!x2 $ y2" or !x2 # y2"3 " 9!x2 $ y2"2

r3 " 3!r2 cos2 ! $ r2 sin2 !"

r " 3!cos2 ! $ sin2 !"

r " 3 cos 2!

77.

y2 " 2x # 1

x2 # y2 " 1 # 2x # x2

$x2 # y2 $ x " 1

r $ r cos ! " 1

r "1

1 $ cos !78.

x2 # 4y $ 4 " 0

x2 # y2 " 4 $ 4y # y2

x2 # y2 " !2 $ y"2

$x2 # y2 # y " 2

r # r sin ! " 2

r "2

1 # sin !©

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. All

right

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serv

ed.


79.

4x2 $ 5y2 $ 36y $ 36 " 0

4x2 # 4y2 " 36 # 36y # 9y2

4!x2 # y2" " !6 # 3y"2

2!±$x2 # y2" " 6 # 3y

2r " 6 # 3r sin !

r!2 $ 3 sin !" " 6

r "6

2 $ 3 sin !80.

2x $ 3y " 6

1 "6

2x $ 3y

r "6r

2x $ 3y

r "6

2!x#r" $ 3!y#r"

r "6

2 cos ! $ 3 sin !

81.

The graph is a circle centered at the origin with radius 7.

x2 # y2 " 49

r2 " 49

x

y

"2"4"8 2 4 6 8"2

"4

"6

"8

2

4

6

8

r " 7 82.

Circle of radius 8 centered at origin

x2 # y2 " 64

r2 " 64

2

42

4

6

10

6 10"6"10 "2

"6

"10

"4

"4x

y r " 8

83.

The graph is the linewhich makes an

angle of withthe positive axis.x-

! " %#4y " x,

y " x

tan ! " tan %4

" 1 "yx

1

2

1

2

3

3"3

"3

"2

"2"1

"1x

y ! "%4

84.

Line through origin making angle of with positive x-axis

%#6

3y $ $3x " 0

yx

" tan ! " tan 7%

6"

$33

x

y

"2"3 1 2 3"1

"2

"3

1

2

3

! "7%

6

85.

Vertical line

x $ 3 " 0

x " 3

r cos ! " 32

1

2 41

3

"3

"2

"2"1

"1x

yr " 3 sec ! 86.

Horizontal line through!0, 2"

y $ 2 " 0

y " 2

r sin ! " 2

1

21

3

4

3"3

"2

"2"1

"1x

y r " 2 csc !

87. True, the distances from the origin are the same. 88. False. For instance when any value of gives the same point.

!r " 0,

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89. (a) where and

where and

Then and Thus,

(b) If the points are on the same line through the origin. In this case,

(c) If the Pythagorean Theorem.

(d) For instance, gives gives (Same!)d ' 2.053.%$3, 7%

6 &, %$4, 4%

3 &d ' 2.053 and%3, %

6&, %4, %

3&!1 $ !2 " 90', d " $r1

2 # r22,

d " $r12 # r2

2 $ 2r1r2 cos!0" " $!r1 $ r2"2 " (r1 $ r2(.!1 " !2,

" $r12 # r2

2 $ 2r1r2 cos!!1 $ !2".

" $r12 # r2

2 $ 2!r1r2 cos !1 cos !2 # r1r2 sin !1 sin !2"

" $!x12 # y1

2" # !x22 # y2

2" $ 2!x1x2 # y1y2"

" $x12 $ 2x1x2 # x2

2 # y12 $ 2y1y2 # y2

2

d " $!x1 $ x2"2 # !y1 $ y2"2

x22 # y2

2 " r22.x1

2 # y12 " r1

2 cos2 !1 # r12 sin2 !1 " r1

2

y2 " r2 sin !2.x2 " r2 cos !2!r2, !2" " !x2, y2"

y1 " r1 sin !1.x1 " r1 cos !1!r1, !1" " !x1, y1"

90. Answers will vary.

91.

C ' 180' $ 30.7' $ 48.2' ' 101.1'

B ' 48.2'

cos B "a2 # c2 $ b2

2ac"

132 # 252 $ 192

2!13"!25" " 0.66615

A ' 30.7'

cos A "b2 # c2 $ a2

2bc"

192 # 252 $ 132

2!19"!25" " 0.86 92.

c "a sin Csin A

' 15.17

C " 180' $ A $ B ' 141.9'

sin B "b sin A

a' 0.2440 ! B ' 14.1'

A " 24', a " 10, b " 6

93.

bsin B

"c

sin C ! b "

c sin Bsin C

"12 sin!86'"

sin!38'" ' 19.44

asin A

"c

sin C ! a "

c sin Asin C

"12 sin!56'"

sin!38'" ' 16.16

B " 180' $ 56' $ 38' " 86' 94.

A " 180' $ B $ C ' 41.9'

sin C " c sin B

b' 0.9214 ! C ' 67.1'

b2 " a2 # c2 $ 2ac cos B ' 885.458 ! b ' 29.76

B " 71', a " 21, c " 29

95.

A " 180' $ B $ C " 119.1'

! B ' 25.9'

b

sin B"

csin C

! sin B "b sin C

c"

4 sin!35'"5.25

c ' 5.25

' 27.57

" 82 # 42 $ 2!8"!4" cos!35'"

c2 " a2 # b2 $ 2ab cos C 96.

a "b sin Asin B

' 53.06

A " 180' $ B $ C ' 66.5'

sin C "c sin B

b' 0.7605 ! C ' 49.5'

B " 64', b " 52, c " 44©

Hou

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ifflin

Com

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. All

right

s re

serv

ed.


97. By Cramer’s Rule,

,

Solution: !2, 3"

y "Dy

D"

$48$16

" 3x "Dx

D"

$32$16

" 2

Dy " ( 5$3

$11$3( " $15 $ 33 " $48

Dx " ($11$3

$71( " $11 $ 21 " $32

D " ( 5$3

$71( " 5 $ 21 " $16


,

Solution: %$526

, 5526&

y "Dy

D"

$55$26

"5526

x "Dx

D" $

526

,

Dy " (34 10$5( " $15 $ 40 " $55

Dx " ($10$5

5$2( " $20 $ !$25" " 5

D " (34 5$2( " $6 $ 20 " $26


, ,

Solution: !0, 0, 0"

c "Dc

D" 0b "

Db

D" 0a "

Da

D" 0

Dc " (321 $21

$3

000( " 0

Db " (321 000

1$3

9( " 0

Da " (000 $21

$3

1$3

9( " 0

D " (321 $21

$3

1$3

9( " 35


Solution: %29589

, 84489

, $67289 &

w "Dw

D"

672$89

"$672

89

v "Dv

D"

$844$89

"84489

,u "Du

D"

$295$89

"29589

,

Dw " (518 7$2$2

1570( " 672

Dv " (518 1570

9$3

1( " $844

Du " (1570

7$2$2

9$3

1( " $295

D " (518 7$2$2

9$3

1( " $89

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Solution: !2, $3, 3 "

z "Dz

D"

$45$15

" 3

y "Dy

D"

45$15

" $3,x "Dx

D"

$30$15

" 2,

Dz " ($125

134

1$2

4( " $45

Dy " ($125

1$2

4

212( " 45

Dx " ( 1$2

4

134

212( " $30

D " ($125

134

212( " $15

102. Cramer’s Rule does not apply because

Use elimination to solve the system.

Let then and

Solution: !$2a # 32, 2a # 1, a"

x1 " $2a # 32.x2 " 2a # 1x3 " a,

)200

010

4$2

0

!!!

310*

$R2 # R1

2R2 # R3

)200

11

$2

2$2

4

!!!

41

$2*"$R1 # R2

$R1 # R3

)222

12

$1

206

!!!

452*"

D " (222 12

$1

206( " 0.

103. Points:

The points are not collinear.

( 46

$2

$3$7$1

111( " $20 ( 0

!4, $3", !6, $7", !$2, $1" 104. Points:

($204

41

$5

111( " 0 ! collinear

!$2, 4", !0, 1", !4, $5"

105. Points:

The points are collinear.

($6$11.5

$4$3

$2.5

111( " 0

!$6, $4", !$1, $3", !1.5, $2.5" 106. Points:

($2.3$0.5

1.5

50

$3

111( " 4.6 ! not collinear

!$2.3, 5", !$0.5, 0", !1.5, $3"

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Section 9.7 Graphs of Polar Equations 845

3. Lemniscate

Section 9.7 Graphs of Polar Equations

! When graphing polar equations:

1. Test for symmetry

(a) Replace by

(b) Polar axis: Replace by

(c) Pole: Replace by

(d) is symmetric with respect to the line

(e) is symmetric with respect to the polar axis.

2. Find the ! values for which is maximum.

3. Find the ! values for which

4. Know the different types of polar graphs.

(a) Limaçons (b) Rose curves, (c) Circles (d) Lemniscates

! You should be able to graph polar equations of the form with your graphing utility. If your utilitydoes not have a polar mode, use

in parametric mode.

y " f !t" sin t

x " f !t" cos t

r " f !!"r " a

r2 " a2 sin 2!r " a sin !r " a sin n!r " a ± b sin !

r2 " a2 cos 2!r " a cos !r " a cos n!r " a ± b cos !

n ! 2

r " 0.#r#

r " f !cos !"! " #$2.r " f !sin !"

!r, # $ !" or !%r, !".!r, !"!r, %!" or !%r, # % !".!r, !"

!r, # % !" or !%r, %!".!r, !"! " #$2:

Vocabulary Check

1. 2. polar axis 3. convex limaçon

4. circle 5. lemniscate 6. cardioid

! "#2

1. is a rose curve.r " 3 cos 2! 2. Cardioid

5. is a rose curve.r " 6 sin 2! 6. Limaçon4. is a circle.r " 3 cos !

7. The graph is symmetric about the line andpasses through Matches (a).!r, !" " !3, 3#$2".

! " #$2, 8. The graph is symmetric about the polar axis andpasses through Matches (c).!r, !" " !3, 0".

9. The graph has four leaves. Matches (c). 10. The graph has three leaves. Matches (d).

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11.

Not an equivalent equation


Polar axis:

Equivalent equation

Pole:



Answer: Symmetric with respect to polar axis

r " 14 % 4 cos !

r " 14 $ 4 cos!# $ !"

%r " 14 $ 4 cos !

r " 14 $ 4 cos !

r " 14 $ 4 cos!%!"

r " 14 % 4 cos !

r " 14 $ 4!cos # cos ! $ sin # sin !"

r " 14 $ 4 cos!# % !"

%r " 14 $ 4 cos !

%r " 14 $ 4 cos!%!"! "#2

:

r " 14 $ 4 cos ! 12.



Polar axis:

Equivalent equation

Pole:



Answer: Symmetric with respect to polar axis

r " %12 cos 3!

r " 12 cos!3!# $ !""

%r " 12 cos 3!

r " 12 cos 3!

r " 12 cos!3!%!""

r " %12 cos 3!

r " 12 cos!3!# % !""

%r " 12 cos 3!

%r " 12 cos!3!%!""! "#

2:

r " 12 cos 3!

13.

Equivalent equation

Polar axis:



%r "4

1 $ sin !

%r "4

1 $ sin!# % !"

r "4

1 % sin !

r "4

1 $ sin!%!"

r "4

1 $ sin !

r "4

1 $ sin # cos ! % cos # sin !

r "4

1 $ sin!# % !"! "#2

:

r "4

1 $ sin !

Pole:



Answer: Symmetric with respect to ! "#2

r "4

1 % sin !

r "4

1 $ sin!# $ !"

%r "4

1 $ sin !

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14.



Polar axis:

Equivalent equation

r "2

1 % cos !

r "2

1 % cos!%!"

r "2

1 $ cos !

r "2

1 % !cos # cos ! $ sin # sin !"

r "2

1 % cos!# % !"

%r "2

1 % cos !

%r "2

1 % cos!%!"! "#2

:

r "2

1 % cos !

Pole:



Answer: Symmetric with respect to the polar axis

r "2

1 $ cos !

r "2

1 % !cos # cos ! % sin # sin !"

r "2

1 % cos!# $ !"

%r "2

1 % cos !

15.

Equivalent equation

Polar axis:



Pole:



Answer: Symmetric with respect to ! "#2

r " %6 sin !

r " 6 sin!# $ !"

%r " 6 sin !

%r " 6 sin !

%r " 6!sin # cos ! % cos # sin !"%r " 6 sin!# % !"

r " %6 sin !

r " 6 sin!%!"

r " 6 sin !

%r " 6 sin!%!"! "#2

:

r " 6 sin ! 16.

Equivalent equation

Polar axis:

Equivalent equation

Pole:

Equivalent equation

Answer: Symmetric with respect to

polar axis and pole

! "#2

,

r " 4 cot !

r " 4 cot!# $ !"

r " 4 cot !

%r " 4 cot!%!"

%r " 4 cot!# % !"

r " 4 cot !

%r " 4 cot!%!"! "#2

:

r " 4 csc ! cos ! " 4 cot !

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17.



Polar axis:



Pole:

Equivalent equation

Answer: Symmetric with respect to pole

r2 " 16 sin 2!

!%r"2 " 16 sin!2!"

r2 " %16 sin 2!

!%r"2 " 16 sin!2!# % !""

r2 " %16 sin 2!

r2 " 16 sin!2!%!""

r2 " %16 sin 2!

r2 " 16 sin!2# % 2!"

r2 " 16 sin!2!# % !""

r2 " %16 sin 2!

!%r"2 " 16 sin!2!%!""! "#2

:

r2 " 16 sin 2! 18.

Equivalent equation

Polar axis:

Equivalent equation

Pole:

Equivalent equation

Answer: Symmetric with respect to

polar axis and pole

! "#2

,

r2 " 25 cos 4!

!%r"2 " 25 cos 4!

r2 " 25 cos 4!

r2 " 25 cos!4!%!""

r2 " 25 cos 4!

!%r"2 " 25 cos!4!%!""! "#2

:

r2 " 25 cos 4!

19.

Maximum: when

! "#2

sin ! " 1

r " 0 when 1 % sin ! " 0

! "3#

2#r# " 20

! "3#

2 or Not possible

sin ! " %1 sin ! " 3

1 % sin ! " 2 or 1 % sin ! " %2

#1 % sin !# " 2

" 10#1 % sin !# " 10!2" " 20

#r# " #10!1 % sin !"# 20.

Maximum:

Zero: r " 0 when ! "2#

3,

4#

3

#r# " 18 when ! " 0

! " 0

cos ! " 1

" 6 $ 12#cos !# " 18

#r# " #6 $ 12 cos !# " #6# $ #12 cos !#

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21.

Maximum: when

! "#

6,

#

2,

5#

6

r " 0 when cos 3! " 0

! " 0, #

3,

2#

3, ##r# " 4

! " 0, #

3,

2#

3, #

cos 3! " ±1

#cos 3!# " 1

#r# " #4 cos 3!# " 4 #cos 3!# " 4 22.

Maximum: when

Zero: when ! " 0, #

2, #,

3#

2, 2#r " 0

! "#

4,

3#

4,

5#

4,

7#

4#r# " 1

#r# " #sin 2!#r " sin 2!

23.

Circle

2 4 6 8

!2

0

r " 5 24.

Line

1 20

2!

! " %5#3 25.

Symmetric with respect to

Circle with radius of

21 30

2!

3$2

! " #$2

r " 3 sin !

26.

Circle

Radius: 1, center:

0

!2

1 3

!1, 0"

r " 2 cos ! 27.

Cardioid

2 40

2!

r " 3!1 % cos !" 28.

Cardioid

0

!2

4 6 8

r " 4!1 $ sin !"

29.

Limaçon

01 2

2!

r " 3 % 4 cos ! 30.

Limaçon with inner loop

0

!2

1 2

r " 1 % 2 sin ! 31.

Limaçon

4 6 8

!2

0

r " 4 $ 5 sin !

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32.

Limaçon

4 6 8 10 12

!2

0

r " 3 $ 6 cos ! 33.

Rose curve

01 65

2!

r " 5 cos 3! 34.

Rose curve

01

2!

r " %sin 5!

35.

Rose curve, four petals

04 5 6

2!

r " 7 sin 2! 36.

Rose curve, five petals

01 2 3 4

2!

r " 3 cos 5! 37.

0 " ! < 2#

18"18

"12

12

r " 8 cos 2!

39.

0 " ! < 2#

18"18

"14

10

r " 2!5 % sin !"38.

0 " ! " 2#

"2

2

3"3

40.

0 " ! " 2#

"12

4

12"12

41.

"6

6

4"14

0 " ! " 2#r " 3 % 6 cos !, 42.

0 " ! " 2#

"6

2

6"6

43.

"4

4

6"6

0 " ! "#2

r "3

sin ! % 2 cos !,

44.

0 " ! " 2#

"4

4

6"6

45.

"2

2

3"3

%2# " ! " 2#r2 " 4 cos 2!, 46.

Graph both functions using

"4

4

6"6

0 " ! " 2#.

r " ±3%sin !

r2 " 9 sin !

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47.

"2

2

3"3

r " 4 sin ! cos2 !, 0 " ! " # 48.

0 " ! " #

"2

2

3"3

49.

0 " ! < 2#

18"18

"10

14

r " 2 csc ! $ 6

50.

0 " ! " 2#

"12

12

18"18

51.

Answers will vary.

"2000

"1400

400

200

r " e2! 52.

Answers will vary.

#15 30

#15

15

r " e!$2

53.

"4

4

6"6

0 " ! < 2#r " 3 % 2 cos !, 54.

0 " ! < 2#

"7

1

6"6

55.

"2

2

3"3

r " 2 cos&3!

2 ', 0 " ! < 4#

56.

0 " ! < 4#

"4

4

6"6

57.

Use and

"1

1

1"1

r2 " %%sin 2!."r1 " %sin 2!!

r2 " sin 2!, 0 " ! <#

2

58.

r "±1%!

0 < ! < &

"1

1

1.5"1.5

59.

is an asymptote.

"4

4

6"6

x " %1

r " 2 % sec !

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60.

The graph has an asymptote at y " 1.

" ±# yy % 1#%3 $ 2y % y2 x " ±%y2!3 $ 2y % y2"

! y % 1"2

x2 "y2!3 $ 2y % y2"

! y % 1"2

x2 $ y2 "4y2

! y % 1"2

!±%x 2$ y 2" "2y

y % 1

!±%x 2 $ y 2" ! y % 1" " 2y

!±%x2 $ y2"! y" " 2y $ !±%x2 $ y2" r!r sin !" " 2r sin ! $ r

r sin ! " 2 sin ! $ 1

"3

5

6"6

r " 2 $ csc ! " 2 $1

sin !

61.

is an asymptote.y " 2

"1

3

3"3

r "2!

62.

"4

4

6"6

63. True. It has five petals. 64. False. For example, let r " cos 3!.

65. Answers will vary.r " cos!5!" $ n cos !, 0 " ! < #;

n " %5

"4

4

6"6

n " %4

"4

4

6"6

n " %3

"4

4

6"6

n " %2

"2

2

3"3

n " %1

"2

2

3"3

n " 0

"2

2

3"3

n " 1

"2

2

3"3

n " 2

"2

2

3"3

n " 3

"4

4

6"6

n " 4

"4

4

6"6

n " 5

"4

4

6"6 ©H

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.


69. (a)

(b)

(c)

(d)

" 4 sin ! cos !

" 2 sin 2!

" 2 sin!2! % 2#"

r " 2 sin(2!! % #")

" %3 cos 2! % sin 2!

" 2 sin&2! %4#

3 '

r " 2 sin*2&! %2#

3 '+ " %4 sin ! cos !

" %2 sin 2!

" 2 sin!2! % #"

r " 2 sin*2&! %#

2'+ " sin 2! % %3 cos 2!

" 2 sin&2! %#

3'

r " 2 sin*2&! %#

6'+ 70. (a)

(b)

–2 –1 1 2

–3

–2

1

x

y

r " 1 % sin&! %#

4'

–2 1 2

–3

–1

1

x

y

r " 1 % sin !

66. The graph of is rotated about the pole through an angle Let be any point on the graph of Then is rotated through the angle and since

it follows that is on the graph of r " f !! % '".!r, ! $ '"r " f !!! $ '" % '" " f !!",',!r, ! $ '"r " f !!".!r, !"'.r " f!!"

67. Use the result of Exercise 66.

(a) Rotation:

Original graph:

Rotated graph:

(b) Rotation:

Original graph:

Rotated graph:

(c) Rotation:

Original graph:

Rotated graph: r " f &sin&! %3#

2 '' " f !cos !"

r " f !sin !"

' "3#

2

r " f !sin!! % #"" " f !%sin !"

r " f !sin !"

' " #

r " f &sin&! %#

2'' " f !%cos !"

r " f !sin !"

' "#

2

68. (a)

(b)

(c)

(d)

r " 2 % cos !

r " 2 % sin&! %3#

2 'r " 2 $ sin !

r " 2 % sin!! % #"

r " 2 $ cos !

r " 2 % sin&! %#

2'

" 2 %%22

!sin ! % cos !"

r " 2 % sin&! %#

4'

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Vocabulary Check1. conic 2. eccentricity, 3. (a) i (b) iii (c) iie


Section 9.8 Polar Equations of Conics

! The graph of a polar equation of the form

or

is a conic, where is the eccentricity and is the distance between the focus (pole) and the directrix.

(a) If the graph is an ellipse.

(b) If the graph is a parabola.

(c) If the graph is a hyperbola.

! Guidelines for finding polar equations of conics:

(a) Horizontal directrix above the pole:

(b) Horizontal directrix below the pole:

(c) Vertical directrix to the right of the pole:

(d) Vertical directrix to the left of the pole: r !ep

1 " e cos #

r !ep

1 $ e cos #

r !ep

1 " e sin #

r !ep

1 $ e sin #

e > 1,

e ! 1,

e < 1,!p!e > 0

r !ep

1 ± e sin #r !

ep1 ± e cos #

71.

Circle

!4

4

6!6

k ! 0

r ! 2 $ k cos #

Convex limaçon

!4

4

6!6

k ! 1

Cardioid

!4

4

8!4

k ! 2


!4

4

8!4

k ! 3

72.

(a)

k ! 1.5: 0 ! # < 4%

!4

4

6!6

r ! 3 sin k #

(b)

k ! 2.5: 0 ! # < 4%

!4

4

6!6

(c) Yes. Answers will vary.

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Section 9.8 Polar Equations of Conics 855

3.

(a) Parabola

(b) Ellipse

(c) Hyperbola!8

4

9!9

a

c

br !

2e1 " e sin #

4. (a) Parabola

(b) Ellipse

(c) Hyperbola

!4

8

9!9

ab

c

5.

parabola

Vertical directrix to left of pole

Matches (b).

e ! 1 "

r !4

1 " cos #6.

ellipse


Matches (c).

e !12

"

r !3

2 " cos #!

3"21 " #1"2$ cos #

8.

hyperbola

Horizontal directrix below the pole.

Matches (e).

e ! 3 "

r !4

1 " 3 sin #9.

hyperbola

Horizontal directrix above the pole.

Matches (d).

e ! 2 "

r !3

1 $ 2 sin #

13.

ellipse

Vertices:

#r, #$ ! %43

, 0&, %45

, %&

e !14

, p ! 4,0

1 2

"2r !

44 " cos #

!1

1 " #1"4$ cos #

1.

(a) Parabola

(b) Ellipse

(c) Hyperbola!4

4

8!4

a

bc

r !2e

1 $ e cos #2. (a) Parabola

(b) Ellipse

(c) Hyperbola

!4

4

4!8

a

bc

7.

ellipse

Vertical directrix to right of pole

Matches (f).

e !12

"

r !3

2 $ cos #!

3"21 $ #1"2$ cos #

10.

parabola

Vertex:

Matches (a).

%2, %2&

e ! 1 "

r !4

1 $ sin #11.

parabola

Vertex: #r, #$ ! #1, %$

e ! 1 "

r !2

1 " cos #12.

Vertex: #1, %"2$

e ! 1 " parabola

r !2

1 $ sin #

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17.

ellipse

Vertices:

%2, %2&, %6,

3%2 &

0

"2

1 2 4 5

e !12

"

r !6

2 $ sin #!

#1"2$#6$1 $ #1"2$ sin #

18.

hyperbola

Vertices: #5, 0$, %"53

, %&e ! 2 "

r !5

"1 $ 2 cos #!

"51 " 2 cos #

19.

hyperbola

Hyperbola

Vertices:

%14

, %&#r, #$ ! %"34

, 0&,0

1 2 3 4 5

"2

e ! 2 "

r !3

4 " 8 cos #!

3"41 " 2 cos #

23.

Hyperbola

!1

1 $ #17"14$ sin #

!3

9

9!9

r !14

14 $ 17 sin #24.

Ellipse

!10

10

15!15

r !12

2 " cos #

14.

Vertices: #r, #$ ! %78

, %2&, %7

6,

3%2 &

e !17

" ellipse

r !7

7 $ sin #!

11 $ #1"7$ sin #

15.

Vertices:

#r, #$ ! %87

, %2&, %8,

3%2 &

01 3 4 5 6

"2e !

34

" ellipse

r !8

4 $ 3 sin #!

21 $ #3"4$ sin #

16.

ellipse

Vertices: #6, 0$, %65

, %&

e !23

"

r !6

3 " 2 cos #!

21 " #2"3$ cos #

20.

hyperbola

Vertices:

#r, #$ ! %56

, %2&, %"

53

, 3%2 &

e ! 3 "

r !10

3 $ 9 sin #!

10"31 $ 3 sin #

21.

Parabola

!4

4

6!6

r !"5

1 " sin #22.

Hyperbola

!2

2

3!3

r !"1

2 $ 4 sin #

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28.

!2

2

3!3

29.

3#3

#2

2 30.!9

!9

9

3

31.

!8

4

9!9

32.

!7

1

4!8

33.

Vertical directrix to the left of the pole

r !1#1$

1 " 1 cos #!

11 " cos #

e ! 1, x ! "1, p ! 1 34.

Horizontal directrix below the pole

r !1#4$

1 " #1$ sin #!

41 " sin #

e ! 1, y ! "4, p ! 4

35.

Horizontal directrix above the pole

r !#1"2$#1$

1 $ #1"2$ sin #!

12 $ sin #

e !12

, y ! 1, p ! 1 36.

Horizontal directrix below pole

!12

4 " 3 sin #r !

(3"4$41 " #3"4$ sin #

e !34

, y ! "4, p ! 4

37.

Vertical directrix to the right of the pole

r !2#1$

1 $ 2 cos #!

21 $ 2 cos #

e ! 2, x ! 1, p ! 1 38.

Vertical directrix to the left of the pole

r !3"2#1$

1 " #3"2$ cos #!

32 " 3 cos #

e !32

, x ! "1, p ! 1

39. Vertex:


r !1#2$

1 " 1 sin #!

21 " sin #

%1, "%

2& " e ! 1, p ! 2 40. Parabola, vertex:


r !ep

1 $ e cos #!

161 $ cos #

#8, 0$e ! 1,

25.

Ellipse

2

2!4

!2

26.

Hyperbola

1!5

!2

2 27.

#!

!

!3

#2 !4

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43. Center:

Vertical directrix to the right of the pole

r !10

3 $ 2 cos #

2 !2p

3 $ 2 cos 0!

2p5

" p ! 5

r !#2"3$p

1 $ #2"3$ cos #!

2p3 $ 2 cos #

#4, %$, c ! 4, a ! 6, e !23

45. Center:


r !20

3 " 2 cos #

20 !2p

3 " 2 ! 2p " p ! 10

r !#2"3$p

1 " #2"3$ cos #!

2p3 " 2 cos #

#8, 0$, c ! 8, a ! 12, e !ca

!23

44. Center:


r !8

3 $ sin #

p ! 8

2 !p

3 $ sin#%"2$

r !#1&3$ p

1 $ #1&3$ sin #!

p3 $ sin #

%1, 3%

2 &, c ! 1, a ! 3, e !13

46. Center:


r !5#9"5$

4 " 5 sin #!

94 " 5 sin #

p !95

1 !5p

4 " 5 sin#3%"2$

r !#5"4$p

1 " #5"4$ sin #!

5p4 " 5 sin #

%5, 3%

2 &, c ! 5, a ! 4, e !54

47. Center:


Substitute the point rather than

in order to get a directrix between the vertices.

r !5#8"5$

3 $ 5 sin #!

83 $ 5 sin #

p !85

1 !5p

3 $ 5 sin#"3%"2$

%"1, 3%2 &%1,

"3%2 &

r !#5"3$p

1 $ #5"3$ sin #!

5p3 $ 5 sin #

%52

, %2&, c !

52

, a !32

, e !53

48. Center:


r !8

3 $ 5 sin #

1 !5p

3 $ 5 sin#%"2$ " p !85

r !#5"3$p

1 $ #5"3$ sin #!

5p3 $ 5 sin #

%52

, %2&, c !

52

, a !32

, e !ca

!53

42. Vertex:

Horizontal directrix above pole

r !1#20$

1 $ 1 sin #!

201 $ sin #

%10, %

2& " e ! 1, p ! 2041. Vertex:


r !1#10$

1 " 1 cos #!

101 " cos #

#5, %$ " e ! 1, p ! 10

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51.

Perihelion distance:

Aphelion distance:

r ! 92.956 ' 106#1 $ 0.0167$ ' 9.4508 ' 107

r ! 92.956 ' 106#1 " 0.0167$ ' 9.1404 ' 107

'9.2930 ' 107

1 " 0.0167 cos #

r !(1 " #0.0167$2)#92.956 ' 106$

1 " 0.0167 cos #52.

Perihelion distance: miles

Aphelion distance: milesa#1 $ e$ ' 4.3381 ' 107

a#1 " e$ ' 2.8585 ' 107

' 3.4462 ' 107

1 " 0.2056 cos #

r !#1 " 0.20562$#35.983 ' 106$

1 " 0.2056 cos #

a ! 35.983 ' 106, e ! 0.2056

53.

Perihelion:

Aphelion:

r ! 77.841 ' 107#1 $ 0.0484$ ' 8.1609 ' 108 km

r ! 77.841 ' 107#1 " 0.0484$ ' 7.4073 ' 108 km

!7.7659 ' 108

1 " 0.0484 cos #

r !#1 " 0.04842$77.841 ' 107

1 " 0.0484 cos #54.

Perihelion distance: km

Aphelion distance: kma#1 $ e$ ' 1.5041 ' 109

a#1 " e$ ' 1.3494 ' 109

' 1.4225 ' 109

1 " 0.0542 cos #

r !#1 " 0.05422$#142.673 ' 107$

1 " 0.0542 cos #

a ! 142.673 ' 107, e ! 0.0542

55. Neptune

Pluto

(a) Neptune:

Pluto:

(b) Neptune: Perihelion: (c)

Aphelion:

Pluto: Perihelion:

Aphelion:

(d) Yes. Pluto is closer to the sun for just a very short time. Pluto was considered the ninth planet because its mean distance from the sun is larger than that of Neptune.

(e) Although the graphs intersect, the orbits do not, and the planets won’t collide.

5.906 ' 109#1 $ 0.2488$ ' 7.3754 ' 109 km

5.906 ' 109#1 " 0.2488$ ' 4.4366 ' 109 km

4.498 ' 109#1 $ 0.0086$ ' 4.5367 ' 109 km!5 # 109 8 # 109

!7 # 109

7 # 1094.498 ' 109#1 " 0.0086$ ' 4.4593 ' 109 km

r !#1 " 0.24882$5.906 ' 109

1 " 0.2488 cos #!

5.5404 ' 109

1 " 0.2488 cos #

r !#1 " 0.00862$4.498 ' 109

1 " 0.0086 cos #!

4.4977 ' 109

1 " 0.0086 cos #

a ! 5.906 ' 109, e ! 0.2488,

a ! 4.498 ' 109, e ! 0.0086,

49. When

Therefore,

Thus, r !ep

1 " e cos #!

#1 " e2$a1 " e cos #

.

a#1 " e2$ ! ep.

a#1 $ e$#1 " e$ ! ep

a#1 $ e$ !ep

1 " e cos 0

# ! 0, r ! c $ a ! ea $ a ! a#1 $ e$. 50. Minimum distance occurs when

Maximum distance occurs when

r !#1 " e2$a

1 " e cos 0!

#1 " e$#1 $ e$a1 " e

! a#1 $ e$

# ! 0.

r !#1 " e2$a

1 " e cos %!

#1 " e$#1 $ e$a1 $ e

! a#1 " e$

# ! %.

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56. (a) Radius of earth miles. Choose

Vertices:

Thus, Thus,

(b) When and the distance from the surface of the earth to the satellite is

(c) When and distance miles.! 38,370# ! 30(, r ' 42,370

15,029 " 4000 ! 11,029 miles.# ! 60(, r ' 15,029

' 7988.11 " 0.937 cos #

.r !ep

1 " e cos #p !

a#1 " e2$e

' 8525.2.

2a !ep

1 " e cos 0$

ep1 " e cos#%$ !

ep1 " e

$ep

1 $ e!

2ep1 " e2

e !ca

!61,340.565,459.5

' 0.937

c ! 65,459.5 " 4119 ! 61,340.5

a !126,800 $ 4119

2! 65,459.5

#126,800, 0$ and #4119, %$

r !ep

1 " e cos #.' 4000

57.

False. The directrix is below the pole.

r !4

"3 " 3 sin #!

"4"31 $ sin #

58.

False. The graph is not an ellipse.

(It is two ellipses.)

r2 !16

9 " 4 cos%# $%4&

59.

For an ellipse, Hence,

r2 !b2

1 " e2 cos2 #.

r2#1 " e2 cos2 #$ ! b2

"r2e2 cos2 # $ r2 ! b2

"r2%ca&

2

cos2 # $ r2 ! b2, e !ca

"r2c2 cos2 # $ r2a2 ! a2b2

b2 " a2 ! "c2.

r2#b2 " a2$ cos2 # $ r2a2 ! a2b2

r2b2 cos2 # $ r2a2 " r2a2 cos2 # ! a2b2

r2 cos2 #

a2 $r2#1 " cos2 #$

b2 ! 1

r2 cos2 #

a2 $r2 sin2 #

b2 ! 1

x2

a2 $y2

b2 ! 1 60.

!"b2

1 " e2 cos2 #

r2 !b2

e2 cos2 # " 1

r2#e2 cos2 # " 1$ ! b2

r2e2 cos2 # " r2 ! b2

r2%ca&

2 cos2 # " r2 ! b2, e !

ca

r2c2 cos2 # " r2a2 ! a2b2

a2 $ b2 ! c2

r2#b2 $ a2$ cos2 # " r2a2 ! a2b2

r2b2 cos2 # " r2a2 $ r2a2 cos2 # ! a2b2

r2 cos2 #

a2 "r2#1 " cos2 #$

b2 ! 1

r2 cos2 #

a2 "r2 sin2 #

b2 ! 1

x2

a2 "y2

b2 ! 1

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61.

r2 !144

1 " #25"169$ cos2 #!

24,336169 " 25 cos2 #

a ! 13, b ! 12, c ! 5, e !513

x2

169$

y2

144! 1

63.

r2 !b2

1 " e2 cos2 #!

161 " #9"25$ cos2 #

!400

25 " 9 cos2 #

a ! 5, b ! 4, c ! 3, e !35

x2

25$

y2

16! 1

64.

r2 !"b2

1 " e2 cos2 #!

"41 " #10"9$ cos2 #

!36

10 cos2 # " 9

a ! 6, b ! 2, c ! *40 ! 2 *10, e !2*10

6!

*103

x2

36"

y2

4! 1

62.

r2 !"16

1 " #25"9$ cos2 #!

14425 cos2 # " 9

a ! 3, b ! 4, c ! 5, e !53

x2

9"

y2

16! 1

65. Center: ,

r2 !"b2

1 " e2 cos2 #!

"91 " #25"16$ cos2 #

!144

25 cos2 # " 16

" b ! 3b2 ! c2 " a2 ! 25 " 16 ! 9

c ! 5, a ! 4, e !54

#x, y$ ! #0, 0$

66. Center: ,

r2 !b2

1 " e2 cos2 #!

91 " #16"25$ cos2 #

!225

25 " 16 cos2 #

" b ! 3b2 ! a2 " c2 ! 25 " 16 ! 9

c ! 4, a ! 5, e !45

#x, y$ ! #0, 0$

67.


(a)

!5

7

9!9

!6

6

9!9

e ! 0.4 " ellipse

r !4

1 " 0.4 cos #

(b)


Graph is reflected in line

Horizontal directrix below pole

rotation counterclockwise90(

r !4

1 " 0.4 sin #

# ! %"2.

r !4

1 $ 0.4 cos #

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68. The lengths of the major andminor axes increase as pincreases.

Example:

r !#0.5$4

1 $ #0.5$ sin #

r !#0.5$2

1 $ #0.5$ sin #

69. Answers will vary. 70.

Circle

x2 $ y2 ! ay $ bx

r2 ! a r sin # $ b r cos #

r ! a sin # $ b cos #

71.

# !%6

$ n%

tan # !1*3

!*33

4*3 tan # " 3 ! 1 72.

x !%3

$ 2n%, 5%3

$ 2n%

cos x !12

6 cos x " 2 ! 1 73.

# !%3

$ n%, 2%3

$ n%

sin # ! ±*32

sin2 # !34

12 sin2 # ! 9

74.

x !%3

$ n%, 2%3

$ n%

sin x !±*3

2

sin2 x !34

csc2 x !43

9 csc2 x " 10 ! 2 75.

x !%2

$ n%

cot x ! 0

2 cot x ! 0

2 cot x ! 5 cos %2

76.

# !%3

$ 2n%, 5%3

$ 2n%

cos # !12

sec # ! 2

*2 sec # ! 2*2

*2 sec # ! 2 csc %4

For Exercises 77–80: sin v ! "1*2

cos v !1*2

,cos u !45

,sin u ! "35

,

77.

!*210

!1

5*2

!45% 1

*2& " %"35&%"

1*2&

cos#u $ v$ ! cos u cos v " sin u sin v 78.

!"7

5*2!

"7*210

! %"35 &% 1

*2& $ %"1*2&%

45&

sin#u $ v$ ! sin u cos v $ sin v cos u

79.

!*210

!1

5*2

! %"35&% 1

*2& " %"1*2&%

45&

sin#u " v$ ! sin u cos v " sin v cos u 80.

!7

5*2!

7*210

! %45&% 1

*2& $ %"35 &%"1

*2&cos#u " v$ ! cos u cos v $ sin u sin v

81. 12C9 ! 220 83. 10P3 ! 72082. 18C16 ! 153 84. 29P2 ! 812

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Review Exercises for Chapter 9

Review Exercises for Chapter 9 863

1.

x2 ! y2 " 25

" !9 ! 16 " !25 " 5

Radius " !"#3 # 0#2 ! "#4 # 0#2 2.

x2 ! y2 " 289

" !64 ! 225 " !289 " 17

Radius " !"8 # 0#2 ! "#15 # 0#2

3.

"x # 2#2 ! " y # 4#2 " 13

Center " $5 ! "#1#2

, 6 ! 2

2 % " "2, 4#

"12!36 ! 16 "

12!52 " !13

Radius "12!"5 # "#1##2 ! "6 # 2#2 4.

"x # 2#2 ! " y ! 1#2 " 32

Center " $#2 ! 62

, 3 # 5

2 % " "2, #1#

"12!64 ! 64 " 4!2

Radius "12!"6 # "#2##2 ! "#5 # 3#2

5.

Center:

Radius: 6

"0, 0#

x2 ! y2 " 36

12

x2 ! 12

y2 " 18 6.

Center:

Radius:2!3

"2!3

3

"0, 0#

x2 ! y2 " 43

34

x2 ! 34

y2 " 1

7.

Center:

Radius: 1

"12, #3

4# "x # 1

2#2! " y ! 3

4#2" 1

16"x # 12#2

! 16" y ! 34#2

" 16

16"x2 # x ! 14# ! 16" y2 ! 3

2 y ! 916# " 3 ! 4 ! 9

16x2 ! 16y2 # 16x ! 24y # 3 " 0

8.

Center:

Radius: 72

"#4, 3#

"x ! 4#2 ! " y # 3#2 " 494

4"x ! 4#2 ! 4" y # 3#2 " 49

4"x2 ! 8x ! 16# ! 4" y2 # 6y ! 9# " #51 ! 64 ! 36

4x2 ! 4y2 ! 32x # 24y ! 51 " 0

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864 Chapter 9 Topics In Analytic Geometry

9.

Center:

Radius: 4!1!2!3!4!6!7 2 3

!2!3!4!5!6

!8

2

x

y"#2, #3#

"x ! 2#2 ! " y ! 3#2 " 16

"x2 ! 4x ! 4# ! " y2 ! 6y ! 9# " 3 ! 4 ! 9 10.

Center:

Radius: 7

!10!14 2 4 6

!4!6

2468

10

14

x

y"#4, 5#

"x ! 4#2 ! " y # 5#2 " 49

"x2 ! 8x ! 16# ! " y2 # 10y ! 25# " 8 ! 16 ! 25

11. intercepts:

intercepts:

No interceptsy-

" y ! 1#2 " #2, impossible

"0 # 3#2 ! " y ! 1#2 " 7y-

"3 ± !6, 0# x " 3 ± !6

x # 3 " ±!6

"x # 3#2 " 6

"x # 3#2 ! "0 ! 1#2 " 7x- 12. intercepts:

No intercepts

intercepts:

"0, 6 ± !2 # y " 6 ± !2

y # 6 " ±!2

" y # 6#2 " 2

"0 ! 5#2 ! " y # 6#2 " 27y-

x-

"x ! 5#2 " #9, impossible

"x ! 5#2 ! "0 # 6#2 " 27x-

13.

Vertex:

Focus:

Directrix: x " #1

"1, 0#

"0, 0#

y2 " 4"1#x, p " 1

–2 2 4 6 8 10

–6

–4

–2

2

4

6

x

y 4x # y2 " 0 14.

Vertex:

Focus:

Directrix: y " 2

"0, #2#

"0, 0#

x2 " 4"#2#y, p " #2–12 –9 –6 6 9 12

–21

–18

–15

–12

–9

–6

3

x

y y " #18 x2

15.

Vertex:

Focus:

Directrix: x " 9

"#9, 0#

x

y

!4!8!12!16!20 4!4

4

12

"0, 0#

y2 " #36x " 4"#9#x, p " #9

12 y2 " #18x

12 y2 ! 18x " 0 16.

Vertex:

Focus:

Directrix: y " # 1128

"0, 1128#

x

y

316

18

18

316

14

516

1128

116

18

316

! ! ! (0, 0)

0, ))

"0, 0#

x2 " 132 y " 4" 1

128#y, p " 1128

8x2 " 14 y

14 y # 8x2 " 0

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17. Vertex:

Focus:

Parabola opens to left.

y2 " #24x

y2 " 4"#6#x

y2 " 4px

"#6, 0#

"0, 0# 18. Vertex:

Focus:

Vertical axis,

"x # 4#2 " #8"y # 2#

"x # 4#2 " 4"#2#"y # 2#

p " #2

"4, 0#

"4, 2#

19. Vertex:

Passes through

Vertical axis

"x ! 6#2 " #9"y # 4#

"x ! 6#2 " 4"#94#"y # 4#

#94 " p

36 " #16p

"0 ! 6#2 " 4p"0 # 4#

"x ! 6#2 " 4p"y # 4#

"0, 0#

"#6, 4# 20. Vertex:

on graph:

"y # 5#2 " 4" 2524#x " 25

6 x

"0 # 5#2 " 4p"6# ! p " 2524

"6, 0#

"y # 5#2 " 4p"x # 0# " 4px

"0, 5#

21.

Focus:

Slope of tangent line:

intercept:

d1

d2

!2

!1

F

(0, )b

!2 2x

y

(2, !2)

"1, 0#x-

y " #2x ! 2

Equation: y ! 2 " #2"x # 2#

b ! 20 # 2

"4

#2" #2

b " 2!12

! b "52

!d1 " d2

d2 "!"2 # 0#2 ! $#2 !12%

2"

52

d1 "12

! b

$0, #12%

x2 " #2y " 4$#12%y, p " #

12

22.

Focus:

Let be the intercept of the tangent line.

intercept: "8, 0#x-

y "14

x # 2

y # 0 "14

"x # 8#

m "#4 # 0#8 # 8

"14

12

! b "172

! b " 8

d2 "!$#8 !12%

2! "#4 # 0#2 "

172

d1 "12

! b

x-"b, 0#

$#12

, 0%

p " #12

4$#12

x% " y2

#2x " y2

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23.

on curve:

if width is meters.8!6x2 " 96 ! x " 4!6 !y " 0

y "#x2 ! 96

8

x2 " 4"#2#"y # 12# " #8y ! 96

16 " 4p"10 # 12# " #8p ! p " #2

"4, 10#

x2 " 4p"y # 12#

24. (a) Parabola:

Vertex:

Passes through

Circle:

Passes through

Radius:

Center:

x2 ! "y ! 4!3 #2 " 64

k " #!48 " #4!3

k2 " 48

16 ! k2 " 64

"±4#2 ! "0 # k#2 " 82

x2 ! "y # k#2 " 82

"0, k#

r " 8

"±4, 0#

x2 " #4"y # 4#

#1 " p

16 " #16p

16 " 4p"0 # 4#

x2 " 4p"y # 4#

"±4, 0#

"0, 4#(b) Parabola:

Circle:

d " #14 x2 # !64 # x2 ! 4 ! 4!3

d " "#14 x2 ! 4# # "!64 # x2 # 4!3 #

x2 ! "y ! 4!3#2" 64 ! y " !64 # x2 # 4!3

x2 " #4"y # 4# ! y " #14 x2 ! 4

x 0 1 2 3 4

d 2.928 2.741 2.182 1.262 0

25.

Center:

Vertices:

Foci:

"!32

"2!3

4

Eccentricity "ca

"0, ±2!3 #

1 !1!3!4 3 4

–3

–2

1

2

3

x

y"0, ±4#

"0, 0#

a " 4, b " 2, c " !16 # 4 " !12 " 2!3

x2

4!

y2

16" 1 26.

Center:

Vertices:

Foci:

Eccentricity "ca

"13

"±1, 0#

!1!2!4 1 2 4!1

!2

!4

1

2

4

x

y"±3, 0#

"0, 0#

a " 3, b " 2!2, c " !9 # 8 " 1

x2

9!

y2

8" 1

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27.

Center:

Vertices:

Foci:

Eccentricity "ca

"!33

"4, #4 ± !3 #"4, #1#, "4, #7#

!1!2 1 2 3 5 6 7 8

!2!3!4!5!6!7!8!9

1x

y"4, #4#

a " 3, b " !6, c " !9 # 6 " !3

"x # 4#2

6!

" y ! 4#2

9" 1 28.

Center:

Vertices:

Foci:

Eccentricity "ca

"!10

4

"#1 ± !10, 3 #"3, 3#, "#5, 3#

!1!2!3!4!5!6 1 2 3 4

12

456789

x

y"#1, 3#

a " 4, b " !6, c " !16 # 6 " !10

"x ! 1#2

16!

" y # 3#2

6" 1

29. (a) (c)

(b) Center:

Vertices:

Foci:

e "ca

"!74

"1, #4 ± !7#"1, 0#, "1, #8#

a " 4, b " 3, c " !16 # 9 " !7

"1, #4#

"x # 1#2

9!

"y ! 4#2

16" 1

16"x # 1#2 ! 9"y ! 4#2 " 144 !1!2!3 1 2 3 4 5

!2

!3

!4

!5

!6

!8

x

y 16"x2 # 2x ! 1# ! 9"y2 ! 8y ! 16# " #16 ! 16 ! 144

30. (a) (c)

(b)

Center:

Vertices:

Foci:

e "!21

5

"#2 ± !21, 3#"3, 3#, "#7, 3#

"#2, 3#

a " 5, b " 2, c " !21

"x ! 2#2

25!

"y # 3#2

4" 1

4"x ! 2#2 ! 25"y # 3#2 " 100

x

y

!2!4!6!8 2 4!2

!4

!6

2

4

6

4"x2 ! 4x ! 4# ! 25"y2 # 6y ! 9# " #141 ! 16 ! 225

31. (a)

—CONTINUED—

"x ! 2#2

1&3!

"y # 7#2

1&8" 1

3"x ! 2#2 ! 8"y # 7#2 " 1

3"x2 ! 4x ! 4# ! 8"y2 # 14y ! 49# " #403 ! 12 ! 392

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31. —CONTINUED—

(b) Center: (c)

Vertices:

Foci:

Eccentricity:ca

"!30&12!3&3

"!10

4

$#2 ±!3012

, 7%$#2 ±

!33

, 7%c2 " a2 # b2 "

13

#18

"524

! c "!3012

a "!33

, b "!24

!1!2!3!4

2

4

6

8

x

y"#2, 7#

32. (a)

(c)

x

y

!1 1 2 3 4

!1

!2

!3

!4

1

'x # "5&2#(2

"5&4# !"y ! 3#2

"1&16# " 1

$x #52%

2! 20"y ! 3#2 "

54

$x2 # 5x !254 % ! 20"y2 ! 6y ! 9# " #185 !

254

! 180

x2 ! 20y2 # 5x ! 120y ! 185 " 0

33. Vertices:

Foci:

x2

25!

y2

9" 1

a " 5, c " 4 ! b " 3

"±4, 0#

"±5, 0#

(b)

Center:

Vertices:

Foci:

e "ca

"!19&4!5&2

"!95

10

$52

±!19

4, #3%

$52

±!52

, #3%$5

2, #3%

a "!52

, b "14

, c "!54

#116

"!19

4

34. Vertices:

Passes through

Vertical major axis

Center:

x2

9&2!

y2

36" 1

b2 "368

"92

4b2 " 1 #

19

"89

22

b2 !22

36" 1

x2

b2 !y2

36" 1

"0, 0#, a " 6

"2, 2#

"0, ±6#

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35. Vertices:

Foci:


Center:

"x # 2#2

25!

y2

21" 1

"x # h#2

a2 !"y # k#2

b2 " 1

b " !25 # 4 " !21

a " 5, c " 2,

"2, 0#

"0, 0#, "4, 0#

"#3, 0#, "7, 0# 36. Vertices:

Foci:

Vertical major axis

Center:

"x # 2#2

3!

"y # 2#2

4" 1

"x # h#2

b2 !"y # k#2

a2 " 1

b " !4 # 1 " !3

a " 2, c " 1,

"2, 2#

"2, 1#, "2, 3#

"2, 0#, "2, 4#

37.

The foci should be placed 3 feet on either side ofthe center and have the same height as the pillars.

c " !a2 # b2 " !25 # 16 " 3b " 4,a " 5, 38.

Longest distance: feet

Shortest distance: feet

Foci:

Distance between foci: feet16!2 ) 22.63

"±8!2, 0#c2 " a2 # b2 " 128

2b " 2"14# " 28

2a " 2"18# " 36

x2

324!

y2

196" 1, a " !324 " 18, b " !196 " 14

39.

Adding,Then

e "ca

) 0.0543.

c " 1.5045 $ 109 # 1.427 $ 109 " 0.0775 $ 109

2a " 2.854 $ 109 ! a " 1.427 $ 109.

a ! c " 1.5045 $ 109

a # c " 1.3495 $ 109 40.

x2

1296!

y2

1241.2" 1

b2 " a2 # c2 " 362 # 7.40162 ) 1241.2

e "ca

" 0.2056 ! c " ae " 7.4016

a "722

" 36

41. (a)

(b)

Center:

Vertices:

Foci:

Eccentricity "ca

"32

"0, ±3#

"0, ±2#

"0, 0#

c " !4 ! 5 " 3

a " 2, b " !5,

y2

4#

x2

5" 1

5y2 # 4x2 " 20 (c)

!1!2!3!4!5 1 2 3 4 5

!3!4!5

1

345

x

y

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42. (a)

x2

"9&4# #y2

"9&4# " 1

x2 # y2 "94

(b)

Center:

Vertices:

Foci:

Eccentricity "ca

"3!2&2

3&2" !2

$±3!2

2, 0%

$±32

, 0%"0, 0#

"!92

"3!2

"3!2

2

c "!94

!94

a "32

, b "32

, (c)

!1!3!4!5 1 3 4 5

!2!3!4!5

12345

x

y

43. (a) (c)

(b) Center:

Vertices:

Foci:

Eccentricity:54

"6, #1#, "#4, #1#

"5, #1#, "#3, #1#

a " 4, b " 3, c " 5"1, #1#,

"x # 1#2

16#

"y ! 1#2

9" 1

9"x # 1#2 # 16"y ! 1#2 " 144

!6 !4 4 62

2

4

6

8

!4

!2

!6

!8

x

y 9"x2 # 2x ! 1# # 16"y2 ! 2y ! 1# " 151 ! 9 # 16

44. (a) (c)

(b) Center:

Vertices:

Foci:

Eccentricity:!29

2

"#1, #3 ± !29#"#1, #1#, "#1, #5#

"#1, #3#, a " 2, b " 5, c " !29

"y ! 3#2

4#

"x ! 1#2

25" 1

25"y ! 3#2 # 4"x ! 1#2 " 100

6 8 10!8 !6 4

!10!8!6!4!2

!12!14

42

6

x

y 25"y2 ! 6y ! 9# # 4"x2 ! 2x ! 1# " #121 ! 225 # 4

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46. (a)

(b) Center: (c)

Vertices:

Foci:

Eccentricity: e " !10

"4 ± !10, 4#"4 ± 1, 4#: "3, 4#, "5, 4#

a " 1, b " 3, c " !10

!2 4 8 10 12!2

2

4

6

8

10

12

x

y"4, 4#

"x # 4#2 #"y # 4#2

9" 1

9"x # 4#2 # "y # 4#2 " 9

9"x2 # 8x ! 16# # "y2 # 8y ! 16# " #119 ! 144 # 16 " 9

47.

x2

16#

y2

20" 1

! b " !20 " 2!5

c2 " a2 ! b2 ! 36 " 16 ! b2

a " 4

x2

a2 #y2

b2 " 1 48. Vertices:

Foci:


Center:

y2

1#

x2

8" 1

b " !9 # 1 " !8a " 1, c " 3,

"0, 0#

"0, ±3#

"0, ±1#

45. (a)

(b) Center: (c)

Vertices:

Foci:

Eccentricity: e "ca

"!505!101

" !5

$#6 ± !5052

, 1%$#6 ± !101

2, 1%

a2 "1012

, b2 " 202, c2 "1012

! 202 "5052

!20!30 10 20 30!10

!20

!30

10

20

30

x

y"#6, 1#

"x ! 6#2

"101&2# #"y # 1#2

202" 1

"y # 1#2 # 4"x ! 6#2 " #202

"y2 # 2y ! 1# # 4"x2 ! 12x ! 36# " #59 ! 1 # 144

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49. Foci:

Center:

Asymptotes:

"x # 4#2

16&5#

y2

64&5" 1

b "8!5

16 " a2 ! "2a#2 " 5a2 ! a "4!5

,

c2 " a2 ! b2

y " ±2"x # 4# ! ba

" 2 ! b " 2a

"4, 0#

"0, 0#, "8, 0# ! c " 4 50. Vertical transverse axis

Center:

5y 2

16#

5"x # 3#2

4" 1

"y # k#2

a2 #"x # h#2

b2 " 1

a2 "165

b2 "45

"2b#2 ! b2 " 4

a2 ! b2 " c2

ab

" 2 ! a " 2b

"3, 0# ! c " 2

51.

72.2 miles north

x " 60 ! y2 " b2$x2

a2 # 1% " "1002 # 46.52#$ 602

46.52 # 1% ) 5211.57 ! y ) 72.2

x2

a2 #y2

b2 " 1

b " !c2 # a2

c " 100

a " 46.5

2a " 93

20406080

100

20 AB

(100, 0)

(60, 0)

x

y

!20

!40!60!80

!100

!60

(!100, 0)

d2 # d1 " 186,000"0.0005#

52. Let the friends be at and you at the origin The sound at is heard 2 seconds after Thus, and Thus, using miles,

the hyperbola is

Now place the center at and determine the second hyperbola.

1. 2.

and

"x # 1#2

"25&64# #y2

"39&64# " 1

b2 " 1 #2564

"3964

c " 1

21

1

A

D

BCx

y

!2 !1

!1

21

1

A

D

BC

y

x!2 !1

!1

2a " DB # AD " 6$11005280% ! a "

58

"1, 0#

x2

"25&576# #y2

"2279&576# " 1.

b2 " c2 # a2 " 2279576 .a " 5

24, c " 22a " CD # BD " 2"11005280# " 5

12.B:CA.C,B

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53.

Ellipse

3"x # 2#2 ! 2"y ! 3#2 " 1

3"x2 # 4x ! 4# ! 2"y2 ! 6y ! 9# " #29 ! 12 ! 18

3x2 ! 2y2 # 12x ! 12y ! 29 " 0

54.

A " C " 4 ! Circle

4x2 ! 4y2 # 4x ! 8y # 11 " 0

55.

Hyperbola

"y ! 1#2

7#

"x ! 1#2

"14&5# " 1

5"x ! 1#2 # 2"y ! 1#2 " #14

5"x2 ! 2x ! 1# # 2"y2 ! 2y ! 1# " #17 ! 5 # 2

5x2 # 2y2 ! 10x # 4y ! 17 " 0 56.

Parabola

A " 0, C " #4, AC " 0

#4y2 ! 5x ! 3y ! 7 " 0

57.

Hyperbola

"x% #2

8#

"y% #2

8" 1

12

"x% #2 #12

"y% #2 " 4

!22

"x% # y% # !22

"x% ! y% # " 4

xy " 4

x "!22

"x% # y% #, y "!22

"x% ! y% #

& "'4

!cot 2& "A # C

B" 0

A " 0, B " 1, C " 0y" x"

!2

!2

!3

2

2 3 4 5

345

x

yxy # 4 " 0

58.

Hyperbola

"x% #2

1&4#

"y% #2

1&6" 1

#4"x% #2 ! 6"y% #2 " #1

12

"x% #2 !12

"y% #2 # x%y% # 5""x% #2 # "y% #2# !12

"x% #2 ! x%y% !12

"y% #2 " #1

*!22

"x% # y% #+2

# 10*!22

"x% # y% #+*!22

"x% ! y% #+ ! *!22

"x% ! y% #+2

! 1 " 0

x "!22

"x% # y% #, y "!22

"x% ! y% #

A " C " 1 ! cot 2& " 0 ! & "'4

2

1

1 2

x"y"

x

yx2 # 10xy ! y2 ! 1 " 0

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59.

Ellipse

"x% #2

3!

"y% #2

2" 1

4"x% #2 ! 6"y% #2 " 12

5*12

"x% #2 # x%y% !12

"y% #2+ # "x% #2 ! "y% #2 ! 5*12

"x% #2 ! x%y% !12

"y% #2+ " 12

5*!22

"x% # y% #+2# 2*!2

2"x% # y% #+*!2

2"x% ! y% #+ ! 5*!2

2"x% ! y% #+2

" 12

5x2 # 2xy ! 5y2 " 12

x "!22

"x% # y% #, y "!22

"x% ! y% #

& "'4

!cot 2& " 0

A " 5, B " #2, C " 5y" x"

!1 1 2 3!2!3!1

!2

!3

1

2

3

x

y5x2 # 2xy ! 5y2 # 12 " 0

60.

Parabola

y% " #4"x% #2 # 8x%

8"x% #2 ! 16x% ! 2y% " 0

2'"x% #2 ! "y% #2 # 2x%y% ( ! 4'"x% #2 # "y% #2( ! 2'"x% #2 ! "y% #2 ! 2x%y% ( ! 7"x% # y% # ! 9"x% ! y% # " 0

! 7!2*!22

"x% # y% #+ ! 9!2*!22

"x% ! y% #+ " 0

4*!22

"x% # y% #+2

! 8*!22

"x% # y% # !22

"x% ! y% #+!4*!22

"x% ! y% #+2

x "!22

"x% # y% #, y "!22

"x% ! y% #x

x"y"

y

!3!4 2 3 4

!4

2

3

4cot 2& "4 # 4

8" 0 ! & "

'4

61. (a)

Parabola

(b)

(c)

!10

!5 2

2

y ""8x # 5# ± !"5 # 8x#2 # 4"16x2 # 10x#

2

y2 ! "5 # 8x#y ! "16x2 # 10x# " 0

B2 # 4AC " "#8#2 # 4"16#"1# " 0 62. (a) Ellipse

(b)

(c)

!4

!6 6

4

y "8x ± !"64x2# # 4"7#"13x2 # 45#

14

7y2 # 8xy ! "13x2 # 45# " 0

B2 # 4AC " #300 ! ©

Hou

ghto

n M

ifflin

Com

pany

. All

right

s re

serv

ed.


63. (a) (c)

Parabola

(b)

y ""2!2 # 2x# ± !"2x # 2!2#2 # 4"x2 ! 2!2x ! 2#

2

y2 ! "2x # 2!2 #y ! "x2 ! 2!2x ! 2# " 0

0!15 0

10B2 # 4AC " "2#2 # 4"1#"1# " 0

64. (a)

(b)

(c)

!4

!6 6

4

y " 5x ± !24x2 # 1

y "10x ± !96x2 # 4

2

y "10x ± !100x2 # 4"x2 ! 1#

2

y2 # 10xy ! "x2 ! 1# " 0

" 96 > 0 ! Hyperbola

B2 # 4AC " 100 # 4

66.

Adding:

If

If impossible.

Answer: "4, 3#, "4, #3#

x " #254 , 9"#25

4 # " 4y2,

y2 " 9 ! y " ±3

4y2 " 9"4# " 36

x " 4:

"x # 4#"4x ! 25# " 0 ! x " 4, #254

4x2 ! 9x # 100 " 0

4x2 ! 9x " 100

9x # 4y2 " 0

4x2 ! 4y2 " 100

65. Adding the equations,

Then:

Solution: "#10, 12#

" y # 12#2 " 0 ! y " 12

y2 # 24y ! 144 " 0

4"100# ! y2 # 560 # 24y ! 304 " 0

x " #10.!24x ! 240 " 0

67.

x

y

!4!8!12 8

!4

4

8

12

16

t 0 1 2 3

x 1 4 7

y 15 11 7 3 #5#1

#2#5#8

#1#2

68.

x

y

!1!2!3 1 2 4 5 6!1

1

2

3

4

5

6

7

8

x " !t, y " 8 # t

t 0 1 2 3 4

x 0 1 2

y 8 7 6 5 4

!3!2

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69.

!20

10 20

20

!1030

30

10

!10

40

x

y

y " 25 "x ! 1# ! 5 " 2

5 x ! 275 , line

t " 15 "x ! 1# !

x " 5t # 1, y " 2t ! 5 70.

!15!20

!20

10 20

20

!10

5!5

15

5

!15

!10x

y

y " 8 # 3"14"x # 1## " #3

4x ! 354

!t " 14"x # 1#

x " 4t ! 1, y " 8 # 3t

71.

4!4 !2 6 8 10 12

12

10

8

6

4

2

!4

x

y

y " 4"x # 2# # 3 " 4x # 11, x " 2

t2 " x # 2 !

x " t2 ! 2, y " 4t2 # 3 72.

!3!4

!4

2 4

4

!2

1!1 3

2

1

!2

!3

3

x

y

y " "14 ex#2

" 116e2x!ex " 4t, t " 1

4 ex

x " ln 4t, y " t2

73.

t " x1&3 ! y "12

x2&3

!1!2!3!4 1

1

2

3

4

5

2 3 4

!2

!3

!1

x

yx " t3, y "12

t2 74.

y "16x2 # 1!t "

4x

!4 12!4

4

20

8!8!12x

yx "4t, y " t2 # 1

75.

!4

!6 6

4

y " x3

t " x3 ! y " t " x3

y " t

x " 3!t 76.

!2

!3 3

2

y " 3!t " 3!x " x1&3

x " t 77.

!4

!6 6

4

y " t "1x

y " t

x "1t

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78.

!4

!6 6

4

y "1t

"1x

x " t 79.

Line

!4

!6 6

4

y " 2"2t# " 2x

y " 4t

x " 2t 80.

!1

!1 5

3

y " 4!x, x " 0

x " "y2#2 " y4, y " 0

t " y2

x " t2, y " !t

81.

3x ! 4y # 11 " 0

y "114

#34

x

!2

!4 8

6 t "x # 1

4 ! y " 2 # 3$x # 1

4 % " 2 #34

x !34

y " 2 # 3t

x " 1 ! 4t

82.

y " "x # 4#2

t " x # 4

!1

!2 10

7x " t ! 4, y " t2 83.

Vertical line: x " 3

y " t

!4

!3 9

4x " 3

84.

y " 2

!1

!6 6

7x " t 85.

x2 ! y2 " 36

x2

36!

y2

36" 1

cos & "x6

, sin & "y6

12!12

!8

8x " 6 cos &, y " 6 sin &

86.

14!4

!4

8"x # 3#2

9!

"y # 2#2

25" 1

cos & "x # 3

3, sin & "

y # 25

x " 3 ! 3 cos &, y " 2 ! 5 sin & 87.

Other answers possible

x " #t, y " #6t ! 2

x " t, y " 6t ! 2

y " 6x ! 2

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88.

Many answers possible

x " #t, y " 10 ! t

x " t, y " 10 # t 89.

Other answers possible

x " t ! 1, y " "t ! 1#2 ! 2 " t2 ! 2t ! 3

x " t, y " t2 ! 2

y " x2 ! 2

90.

Many answers possible

x " #t, y " #2t3 # 5t

x " t, y " 2t3 ! 5t 91.

or x " t, y " 5

y " y1 ! t"y2 # y1# " 5 ! t"0 # 0# " 5

x " x1 ! t"x2 # x1# " 3 ! t"8 # 3# " 5t ! 3

92.

or x " 2, y " t

" #1 ! t"4 # "#1## " #1 ! 5t

y " y1 ! t"y2 # y1#

x " x1 ! t"x2 # x1# " 2 ! t"2 # 2# " 2 93.

y " y1 ! t"y2 # y1# " 6 ! t"0 # 6# " #6t ! 6

" #1 ! t'10 # "#1#( " 11t # 1

x " x1 ! t"x2 # x1#

94.

or x " 5t, y " 12t

y " y1 ! t"y2 # y1# " 0 ! t"6 # 0# " 6t

x " x1 ! t"x2 # x1# " 0 ! t" 52 # 0# " 5

2t 95. is on the curve:

Hence, v0 ) 900.82"2.024# ) 54.23 ft&sec.

16t2 " 3 !0.57"90#

0.82 ! t ) 2.024

4 " 7 ! 0.57* 900.82t+t # 16t2 !

90 " 0.82v0 t ! v0 "90

0.82t

"90, 4#

96. From Exercise 95,

" 7 ! 30.91t # 16t2

y " 7 ! 0.57"54.23#t # 16t2

x " 0.82"54.23#t " 44.47t

v0 " 54.23. 97. From Exercise 96:

The maximum height is approxi-mately 21.9 feet for t ) 0.97.

00 100

25

98. From Exercise 95,seconds.t ) 2.024

99.

$1, #7'4 %, $#1,

5'4 %, $#1, #

3'4 %

01 2 3

1,4#( (

#2

$1, '4% 100.

$5, #4'3 %$#5,

5'3 %, $5,

2'3 %,

0

#2

!5,3#( (!

1 2 3 4 5

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101.

$#2, '6%, $2,

7'6 %, $2, #

5'6 %

01 2 3

#2

# ))!2, 611!

"r, &# " $#2, #11'

6 % 102.

$#1, #'6%, $#1,

11'6 %, $1, #

7'6 %

0

#)( ,

1 2 3

56

#2

1

"r, &# " $1, 5'6 %

103.

$!5, 2'3 %, $#!5, #

'3%, $#!5,

5'3 %

01 2 3

5, !3

4( (

#2

#

$!5, #4'3 % 104.

"r, &# " $!10, #5'4 %, $#!10, #

'4%, $#!10,

7'4 %

01 2 3 4

10,( )4#3

#2

105.

01 2 3

#2

5, !6

7#( (

"x, y# " $#5!3

2,

52%

"r, &# " $5, #7'6 % 106.

"x, y# " $#4 cos 2'3

, #4 sin 2'3 % " "2, #2!3 #

"r, &# " $#4, 2'3 %

0

#2

!4,3

2#( (1 2 3 4 5

107.

"x, y# " "1, !3 #

y " r sin & " 2$!32 % " !3

x " r cos & " 2$12% " 1

01 2 3 4

#2

# ))2, 35!

$2, #5'3 %

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108.

01 2 3

#2

# ))!1, 611

"x, y# " $#!32

, 12%

y " r sin & " #1$#12%

x " r cos & " #1$!32 %

$#1, 11'

6 % 109.

1 2 30

#2

3,43#( (

"x, y# " $3 cos 3'4

, 3 sin 3'4 % " $#3!2

2,

3!22 %

"r, &# " $3, 3'4 %

110. the origin

0,2#( (

0

#2

1 2 3 4 5

"x, y# " "0, 0#

"r, &# " $0, '2%, 111.

!3 3 6 9 12

!12

!9

!6

!3

3

x

y

(0, !9)

$#9, '2%"r, &# " $9,

3'2 %,

"x, y# " "0, #9#

112.

or

(radians)"5, 2.214#, "#5, 5.356#

"5, 126.87(#, "#5, 306.87(#

r " 5, tan & "#43

x

y

!1!3 !2!4 1 32 4!1

!3

!2

!4

1

3

2

4(!3, 4)

"x, y# " "#3, 4# 113.

x

y

(5, !5)

!1!2 1 2 3 4 5 6!1

!2

!3

!4

!5

!6

1

2

$#5!2, 3'4 %"r, &# " $5!2,

7'4 %,

"x, y# " "5, #5#

114.

Third quadrant,

"r, &# " $2!3, 7'6 %, $#2!3,

'6%

r " 2!3!r2 " "#3#2 ! 3 " 12

& "7'6

x

y

!1!3 !2!4 1 32 4!1

!3

!2

!4

1

3

2

4

!3, ! 3( (

"x, y# " "#3, #!3 #

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129.

liney " #!33

x,

tan & "yx

" #1!3

& "5'6

130.

y " !3x

tan & "yx

" !3

& "4'3

119.

r2 " 5 csc & ) sec &

"r cos &#"r sin &# " 5

xy " 5 120.

r2 " #2 sec & csc &

"r cos &#"r sin &# " #2

xy " #2

121.

r2 "1

3 cos2 & ! 1

r2 '3 cos2 & ! 1( " 1

4r2 cos2 & ! r2"1 # cos2 &# " 1

4"r cos &#2 ! "r sin &#2 " 1

4x2 ! y " 1 122.

r2 "1

2 ! sin2 &

r2"2 ! sin2 &# " 1

r2"2"1 # sin2 &# ! 3 sin2 &# " 1

r2"2 cos2 & ! 3 sin2 &# " 1

2"r cos &#2 ! 3"r sin &#2 " 1

2x2 ! 3y2 " 1

123.

Circle

x2 ! y2 " 52 " 25

r " 5 124.

Circle

x2 ! y2 " 144

r " 12 125.

x2 ! y2 " 3x

r2 " 3r cos &

r " 3 cos &

126.

x2 ! y2 " 8y

r2 " 8r sin &

r " 8 sin & 127.

"x2 ! y2#2 # x2 ! y2 " 0

"x2 ! y2#2 " x2 ! y2 # 2y2

r4 " r2 # 2r2 sin2 &

r2 " 1 # 2 sin2 &

r2 " cos 2&

128.

"x2 ! y2#3 " y2

"x2 ! y2#3&2 " y or

r3 " r sin &

r2 " sin &

131. circle

4 620

#2

r " 5, 132. circle

1 2 40

#2

r " 3, 133. axis

2 310

#2

y-& "'2

,

115.

r " 3

r2 " 9

x2 ! y2 " 9 116.

r " 2!5

r2 " 20

x2 ! y2 " 20 117.

r " 4 cos &

r2 # 4r cos & " 0

x2 ! y2 # 4x " 0 118.

r " 6 sin &

r2 " 6r sin &

x2 ! y2 " 6y

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134. line

1 4320

#2

& " #5'6

, 135. circle

064321

#2

r " 5 cos &, 136. circle

1 320

#2

r " 2 sin &,

137.

Dimpled limaçon

Symmetric with respect to polar axis

is maximum at

(No zeros)r * 0

"r, &# " "9, 0#& " 0:r 0

#2

2 4 6 8 1210

r " 5 ! 4 cos &

138.


Symmetric with respect to

is a maximum at

when & ) 3.394, 6.031!sin & " #14

!4 sin & " #1r " 0

& "'2

: $5, '2%,r,

& "'2

01 2 3 4

#2

r " 1 ! 4 sin &

139.

Limaçon with loop

Symmetry: line

Maximum value: when

Zeros: when $sin & "35%& ) 0.6435, 2.4981r " 0

& "3'2,r, " 8,r,-

& "'2

01 2 3 5

#2

r " 3 # 5 sin &

140.


Symmetry: polar axis

Maximum: when

Zero: when cos & " 13 ! & ) 1.231, 5.052r " 0

& " ',r, " 80

#2

2

r " 2 # 6 cos &

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141.

Four-leaved rose

Symmetric with respect to polar axis, and pole

The value of is a maximum (3) at

for & "'4

, 3'4

, 5'4

, 7'4

r " 0

'2

, ', 3'2

.& " 0,,r,

& "'2

,0

4

#2

r " #3 cos 2&, 0 # & # 2'

142.

Five-leaved rose

Symmetric with respect to polar axis

is maximum value of 1 at

for & "'10

!2n'10

, n " 0, 1, 2, . . .r " 0

& "n'5

, n " 0, 1, 2, . . .,r,0

1

#2

r " cos 5&

143.

Lemniscate

Symmetry with respect to pole

Maximum value: when

Zeros: when & " 0, '2

, ', 3'2

r " 0

& "'4

, 5'4

!5,r,-0

1 2 3

#2

r2 " 5 sin 2&

144.

Lemniscate

Symmetry: Pole, polar axis, and line

Maximum: when

Zeros: when & "'4

, 3'4

, 5'4

, 7'4

r " 0

& " 0, ', 2',r, " 1

& "'2 0

1

#2

r2 " cos 2&

145.

Parabola

!2

6

6!6

e " 1

r "2

1 # sin &146.

Hyperbola symmetric with and havingvertices at and

!1

3

3!3

"#1, 3'&2#"1&3, '&2#& " '&2

r "1

1 ! 2 sin &, e " 2

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147.

Ellipse

e "35

"4&5

1 # "3&5# cos &!2

2

3!3

r "4

5 # 3 cos &148.

Hyperbola

!4

4

6!6

"e " 4#

r "6

#1 ! 4 cos &"

#61 # 4 cos &

149.

Ellipse

e "13

"5&6

1 ! "1&3# sin &!2

2

3!3

r "5

6 ! 2 sin &150.

Parabola

!4

4

8!4

"e " 1#

r "3

4 # 4 cos &"

3&41 # cos &

151.

Vertical directrix: x " #4

r "4

1 # cos &

e " 1 152. Parabola:

Vertex:

Focus:

r "4

1 ! sin &

"0, 0# ! p " 4

$2, '

2%

r "ep

1 ! e sin &, e " 1

153. Ellipse:

Vertices:

One focus:

r ""2&3#"5&2#

1 # "2&3# cos &"

5&31 # "2&3# cos &

"5

3 # 2 cos &

5 "2&3 p

1 # "2&3# cos 0 ! p "

52

e "ca

"23

,

"0, 0# ! c " 2

"5, 0#, "1, '# ! a " 3

r "ep

1 # e cos &

154. Hyperbola:

Vertices:

One focus:

"7

3 ! 4 cos &"

7&31 ! "4&3# cos &

r ""4&3#"7&4#

1 ! "4&3# cos &

1 "4&3 p

1 ! "4&3# cos 0 ! p "

74

e "ca

"43

,

"0, 0# ! c " 4

"1, 0#, "7, 0# ! a " 3

r "ep

1 ! e cos &

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155.

Use

Perihelion: astronomical units

Aphelion: astronomical units1.512

1 # 0.093) 1.667

1.5121 ! 0.093

) 1.383

r "1.512

1 # 0.093 cos &

2a "0.093p

1 # 0.093 cos 0!

0.093p1 # 0.093 cos '

" 0.1876p " 3.05 ! p ) 16.258, ep ) 1.512

r "ep

1 # e cos &.

e " 0.093

156. Use (horizontal directrix below pole).

(parabola)

When

When distance is approximately 6,430,781 miles.& " #'3

,

r "12,000,0001 # sin &

r "p

1 # sin$#'2 %

"p2

" 6,000,000 ! p " 12,000,000

& "#'

2, r " 6,000,000.

e " 1

r "ep

1 # e sin &

157. False. The term is notsecond degree.

y4- 158. False. There are many setspossible. For example,

x " 3t, y " 3 # 6t.

x " t, y " 3 # 2t

159. (a) Vertical translation

(b) Horizontal translation

(c) Reflection in the axis

(d) Parabola opens more slowly.

y-

160. (a) Major axis horizontal

(b) Circle

(c) Ellipse is flatter.

(d) Horizontal translation

161. The number must be less than 5. The ellipsebecomes more circular and approaches a circle of radius 5.

b

162. The orientation of the graph would be reversed. 163. (a) The speed would double.

(b) The elliptical orbit would be flatter. The lengthof the major axis is greater.

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Chapter 9 Practice Test

1. Find the vertex, focus and directrix of the parabola x2 ! 6x ! 4y " 1 # 0.

2. Find an equation of the parabola with its vertex at and focus at !2, !6".!2, !5"

3. Find the center, foci, vertices, and eccentricity of the ellipse x2 " 4y2 ! 2x " 32y " 61 # 0.

4. Find an equation of the ellipse with vertices and eccentricity e # 12.!0, ±6"

5. Find the center, vertices, foci, and asymptotes of the hyperbola 16y2 ! x2 ! 6x ! 128y " 231 # 0.

6. Find an equation of the hyperbola with vertices at and foci at !±5, 2".!±3, 2"

7. Rotate the axes to eliminate the term. Sketch the graph of the resulting equation, showing both sets of axes.

5x2 " 2xy " 5y2 ! 10 # 0

xy-

8. Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or hyperbola.

(a) (b) x2 " 4xy " 4y2 ! x ! y " 17 # 06x2 ! 2xy " y2 # 0

For Exercises 9 and 10, eliminate the parameter and write the corresponding rectangular equation.

9. x # 3 ! 2 sin $, y # 1 " 5 cos $ 10. x # e2t, y # e4t

11. Convert the polar point to rectangular coordinates.!#2, !3%"$4"

12. Convert the rectangular point to polar coordinates.!#3, !1"

13. Convert the rectangular equation to polar form.4x ! 3y # 12

14. Convert the polar equation to rectangular form.r # 5 cos $

15. Sketch the graph of r # 1 ! cos $.

16. Sketch the graph of r # 5 sin 2$.

17. Sketch the graph of r #3

6 ! cos $.

18. Find a polar equation of the parabola with its vertex at and focus at !0, 0".!6, %$2"

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CHAPTER 9 Topics in Analytic Geometry - crunchy...

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