of 17
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Copyright 2007 Pearson Education, Inc. Slide 6-1
Chapter 6: Analytic Geometry
6.1 Circles and Parabolas
6.2 Ellipses and Hyperbolas
6.3 Summary of the Conic Sections
6.4 Parametric Equations
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Copyright 2007 Pearson Education, Inc. Slide 6-2
6.1 Circles and Parabolas
Conic Sections
Parabolas, circles, ellipses, hyperbolas
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Copyright 2007 Pearson Education, Inc. Slide 6-3
A circle with center (h, k) and radius rhas lengthto some point (x,y) on
the circle.
Squaring both sides yields the center-radius
form of the equation of a circle.
6.1 Circles
A circle is a set of points in a plane that are equidistantfrom a fixed point. The distance is called the radius of
the circle, and the fixed point is called the center.
22 )()( kyhxr !
22)()( kyhxr
2!
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Copyright 2007 Pearson Education, Inc. Slide 6-4
6.1 Finding the Equation of a Circle
Example Find the center-radius form of the equationof a circle with radius 6 and center (3, 4). Graph the
circle and give the domain and range of the relation.
Solution Substitute h = 3, k= 4, and r= 6 into the
equation of a circle.
22
222
)4()3(36
)4())3((6
!
!
yx
yx
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Copyright 2007 Pearson Education, Inc. Slide 6-5
6.1 Graphing Circles with the Graphing
Calculator
Example Use the graphing calculator to graph thecircle in a square viewing window.
Solution
922 ! yx
.9and9Let
9
9
9
2
2
2
1
2
22
22
xyxy
xy
xy
yx
!!
s!
!
!
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Copyright 2007 Pearson Education, Inc. Slide 6-6
6.1 Graphing Circles with the Graphing
Calculator
TECHNO OGY NOTES:
Graphs in a nondecimalwindow may not be connected
Graphs in a rectangular (non-square) window look like
an ellipse
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Copyright 2007 Pearson Education, Inc. Slide 6-7
6.1 Finding the Center and Radius of a
Circle
Example Find the center and radius of the circlewith equation
Solution Our goal is to obtain an equivalent
equation of the formWe complete the square in bothx andy.
2 26 10 25 0. x x y y !
.)()(
222
kyhxr!
2 2
2 2
2 2
2 2 2
6 10 25
( 6 9) ( 10 25) 25 9 25
( 3) ( 5) 9
( 3) ( 2) 3
x x y y
x x y y
x y
x y
!
!
!
!
The circle has center (3, 2) with radius 3.
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Copyright 2007 Pearson Education, Inc. Slide 6-8
6.1 Equations and Graphs of Parabolas
For example, let the directrix be the liney = c and
the focus be the pointFwith coordinates (0, c).
A parabola is a set of points in a plane equidistant
from a fixed point and a fixed line. The fixed point
is called the focus, and the fixed line, the directrix.
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Copyright 2007 Pearson Education, Inc. Slide 6-9
6.1 Equations and Graphs of Parabolas
To get the equation of the set of points that are the
same distance from the liney = c and the point
(0, c), choose a pointP(x,y) on the parabola. The
distance from the focus,F, toP, and the point on
the directrix,D, toP, must have the same length.
cyx
cycycycyx
cycycycyx
cyxxcyx
DPdFPd
4
22
22
))(()()()0(
),(),(
2
22222
22222
2222
!
!
!
!
!
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Copyright 2007 Pearson Education, Inc. Slide 6-10
6.1 Parabola with a Vertical Axis
The focal chordthrough the focus and perpendicular to theaxis of symmetry of a parabola has length |4c|.
ety =c and solve forx.
The endpoints of the chord are (s x, c), so the length is |4c|.
The parabola with focus (0, c) and directrixy = c has
equationx2 = 4cy. The parabola has vertical axisx = 0,
opens upward ifc > 0, and opens downward ifc < 0.
ccxcx
cyx
2or24
4
22
2
s!!
!
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Copyright 2007 Pearson Education, Inc. Slide 6-11
6.1 Parabola with a Horizontal Axis
Note: a parabola with a horizontal axis is not a function.
The graph can be obtained using a graphing calculator by
solvingy2 = 4cx fory:
et and graph each half of the
parabola.
The parabola with focus (c, 0) and directrixx = c
has equationy2 = 4cx. The parabola has horizontal
axisy = 0, opens to the right ifc > 0, and to the left
ifc < 0.
.2 cxy s!
cxycxy 2and2 21 !!
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Copyright 2007 Pearson Education, Inc. Slide 6-12
6.1 Determining Information about
Parabolas from Equations
Example Find the focus, directrix, vertex, and axis
of each parabola.
(a)
Solution(a)
xyyx 28(b)8 22 !!
2
84
!
!
c
c
Since thex-term is squared, theparabola is vertical, with focusat (0, c) = (0, 2) and directrix
y = 2. The vertex is (0, 0), andthe axis is they-axis.
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Copyright 2007 Pearson Education, Inc. Slide 6-13
6.1 Determining Information about
Parabolas from Equations
(b)
The parabola is horizontal,with focus (7, 0), directrix
x = 7, vertex (0, 0), and
x-axis as axis of the parabola.
Since c is negative, the graphopens to the left.
7284
!
!
cc
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Copyright 2007 Pearson Education, Inc. Slide 6-14
6.1 Writing Equations of Parabolas
Example Write an equation for the parabola withvertex (1, 3) and focus (1, 3).
Solution Focus lies left of the vertex implies the
parabola has- a horizontal axis, and
- opens to the left.
Distance between vertex and
focus is 1(1)=
2, so c=
2.
)1(8)3(
)1)(2(4)3(2
2
!
!
xy
xy
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Copyright 2007 Pearson Education, Inc. Slide 6-15
6.1 An Application of Parabolas
Example Signals coming inparallel to the axis of a parabolic
reflector are reflected to the focus,
thus concentrating the signal.
The Parkes radio telescope has a
parabolic dish shape with diameter 210 feet and depth 32 feet.
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Copyright 2007 Pearson Education, Inc. Slide 6-16
6.1 An Application of Parabolas
(a) Determine the equation describing the cross section.(b) The receiver must be placed at the focus of the parabola.
How far from the vertex of the parabolic dish should the
receiver be placed?
Solution(a) The parabola will have the form y =ax2 (vertex at the
origin) and pass through the point ).32,105(32,2210 !
.025,11
32
bydescribedbecansectioncrosstheso,025,11
32
105
32
)105(32
2
2
2
xy
a
a
!
!!
!
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Copyright 2007 Pearson Education, Inc. Slide 6-17
6.1 An Application of Parabolas
(b) Since
The receiver should be placed at (0, 86.1), or
86.1 feet above the vertex.
,025,11
32 2xy !
.1.86
128
025,11
32
025,114
14
}!
!
!
c
c
ac
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