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Homework, Page 673

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Homework, Page 673. Using the point P ( x, y ) and the rotation information, find the coordinates of P in the rotated x’y’ coordinate system. 33. Homework, Page 673. - PowerPoint PPT Presentation
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 1 Homework, Page 673 Using the point P(x, y) and the rotation information, find the coordinates of P in the rotated x’y’ coordinate system. 33. , 2,5 , 4 Pxy , 2,5 , 4 Pxy cos sin x x y 2 2 2 5 2 2 22 52 2 2 32 2 cos sin y y x 2 2 5 2 2 2 52 22 2 2 72 2 3272 , , 2 2 Pxy
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Page 1: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 1

Homework, Page 673Using the point P(x, y) and the rotation information, find the coordinates of P in the rotated x’y’ coordinate system.

33. , 2,5 , 4P x y

, 2,5 , 4P x y

cos sinx x y 2 22 5

2 2

2 2 5 2

2 2

3 2

2

cos siny y x 2 25 2

2 2

5 2 2 2

2 2

7 2

2

3 2 7 2, ,

2 2P x y

Page 2: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 2

Homework, Page 673Identify the type of conic, and rotate the coordinate system to eliminate the xy-term. Write and graph the transformed equation.

37. 8xy

8xy

x

y

cos sin sin cos 8x y x y 2 2 2 2cos sin cos sin sin cos 8x x y x y y

45 2 2 2 2cos sin cos sin 8x y x y

2 2 2 2 2 2 2 28

2 2 2 2 2 2x y x y

2 2 18

2x y

2 2

116 16

x y

Page 3: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 3

Homework, Page 673Identify the type of conic, solve for y, and graph the conic. Approximate the angle of rotation needed to eliminate the xy-term.

41. 2 216 20 9 40 0x xy y

2 4B AC 220 4 16 9 400 576 0 ellipse

2 29 20 16 40 0y x y x

2 220 20 4 9 16 40

2 9

x x xy

220 176 1440

18

x xy

cot 2A C

B 12 tan

B

A C

1 20

tan16 9

70.710

Page 4: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 4

Homework, Page 673Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola.

45. 2 29 6 7 5 0x xy y x y

2 4B AC 26 4 9 1 36 36 0 Parabola

Page 5: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 5

Homework, Page 673Use the discriminant to decide whether the equation represents a parabola, an ellipse, or a hyperbola.

49. 2 23 22 0x y y

2 4B AC 20 4 1 3 12 0 Hyperbola

Page 6: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 6

Homework, Page 67353. Find the center, vertices, and foci of the hyperbola in the original coordinate system.

2 9 0xy

2 2 2 2

2 2 2 2

2 2

2 2

2 9 0 2 cos sin sin cos 9

2 cos sin sin cos 9

2 cos sin cos sin sin cos 9

2 cos45 sin 45 cos 45 sin 45 9

2 2 2 22

2 2 2 2

xy x y x y

x y x y

x x y x y y

x y x y

x y x y

2 2

2 2

9

9 1 0,0 , 3 2,0 , 3,09 9

x yx y C F V

Page 7: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 7

Homework, Page 67353. Find the center, vertices, and foci of the hyperbola in the original coordinate system.

2 9 0xy

0,0 , 3 2,0 , 3,0

3 2 cos45 0sin 45 3 2 sin 45 0cos45 3, 3

3 2 3 23cos45 0sin 45 3sin 45 0cos45 ,

2 2

3 2 3 20,0 , 3, 3 , ,

2 2

C F V

C F V

Page 8: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 8

Homework, Page 67357.

True, because there is no xy term to cause a rotation.

2 2The graph of the equation 0 A and C not both zero

has a focal axis aligned with the coordinate axes. Justify your answer.

Ax Cy Dx Ey F

Page 9: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 9

Homework, Page 67361.

A. (1±4, –2)

B. (1±3, –2)

C. (4±1, 3)

D. (4±2, 3)

E. (1, –2±3)

2 2The vertices of 9 16 18 64 7 0 are:x y x y

2 2

2 2

2 2

2 2

2 2

2 2

2 2

9 16 18 64 71 0

9 18 16 64 71

9 2 16 4 71

9 2 1 16 4 4 71 9 64

9 1 16 2 144

9 1 16 2 144

144 144 144

1 21

16 91, 2 , 1 4, 2

x y x y

x x y y

x x y y

x x y y

x y

x y

x y

C F

Page 10: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

8.5

Polar Equations of Conics

Page 11: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 11

Quick Review

2

2 2

2 2

1. Solve for . (4, ) ( , )

2. Solve for . (3, 5 /3)=( 3, ), 2 2

3. Find the focus and the directrix of the parabola.

12

Find the focus and the vertices of the conic.

4. 116 9

5. 9 16

r r

x y

x y

x y

1

Page 12: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 12

Quick Review Solutions

2

1. Solve for . (4, ) ( , )

2. Solve for . (3, 5 /3)=( 3, ), 2 2

3. Find the focus and the directrix of the parabola.

12

4

4 / 3

(0,3

Find the focus and the vertices of the conic.

)

4

; 3

r r

x y y

2 2

2 2

. 1 16 9

5.

( 5,0); ( 4,0)

(0, 7); 1 (0, 4) 9 16

x y

x y

Page 13: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 13

What you’ll learn about

Eccentricity Revisited Writing Polar Equations for Conics Analyzing Polar Equations of Conics Orbits Revisited

… and whyYou will learn the approach to conics used by astronomers.

Page 14: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 14

Focus-Directrix Definition Conic Section

A conic section is the set of all points in a plane whose distances from a particular point (the focus) and a particular line (the directrix) in the plane have a constant ratio. (We assume that the focus does not lie on the directrix.)

Page 15: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 15

Focus-Directrix Eccentricity RelationshipIf is a point of a conic section, is the conic's focus, and is the

point of the directrix closest to , then and ,

where is a constant and the eccentricity of the conic.

Moreo

P F D

PFP e PF e PD

PDe

ver, the conic is

a hyperbola if 1,

a parabola if 1,

an ellipse if 1.

e

e

e

Page 16: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 16

A Conic Section in the Polar Plane

Page 17: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 17

Three Types of Conics for r = ke/(1+ecosθ)

Page 18: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 18

Polar Equations for Conics

Page 19: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 19

Example Writing Polar Equations of Conics

Given that the focus is at the pole, write a polar equation for the conic

with eccentricity 4/5 and directrix 3.x

Page 20: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 20

Example Identifying Conics from Their Polar Equations

Determine the eccentricity, the type of conic, and the directrix.

6

3 2cosr

Page 21: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 21

Example Matching Graphs of Conics with Their Polar Equations

Match the polar equation with its graph and identify the viewing window.

9

5 3sinr

Page 22: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 22

Example Finding Polar Equations of Conics

Find a polar equation for the ellipse with a focus at the pole and

the given polar coordinates as the endpoints of the major axis.

1.5,0 and 1,

Page 23: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 23

Example Finding Polar Equations of Conics

Find a polar equation for the hyperbola with a focus at the pole and

the given polar coordinates as the endpoints of its transverse axis.

3,0 and 1.5,

Page 24: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 24

Homework

Homework #23 Review Section 8.5 Page 682, Exercises: 1 – 29(EOO) Quiz next time

Page 25: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 25

Semimajor Axes and Eccentricities of the Planets

Page 26: Homework, Page 673

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8- 26

Ellipse with Eccentricity e and Semimajor Axis a

21

1 cos

a er

e


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