Chapter Solutions

CHAPTER 9
Topics 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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CHAPTER 9
Topics in Analytic Geometry
Section 9.1
Circles and Parabolas
■ A parabola is the set of all points x, y 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 h, k and
(a) Vertical axis x h and directrix y k p is
(x h) 2 4p( y k), p 0.
(b) Horizontal axis y k and directrix x h p is
(y k) 2 4p(x h), p 0.
■ The tangent line to a parabola at a point P makes equal angles with
(a) the line through P and the focus.
(b) the axis of the parabola.
Vocabulary Check
1. conic section 2. locus 3. circle, center
4. parabola, directrix, focus 5. vertex 6. axis
7. tangent
1.
x 2 y 2 18 2 2.
x 2 y 2 18
x 2 y 2 42 2
x 2 y 2 32
3.
5.
7.
Radius 3 1 2 7 0 2 4. Radius 6 2 2 3 4 2
4 49 53
64 49 113
x h 2 y k 2 r 2
x h 2 y k 2 r 2
x 3 2 y 7 2 53
x 6 2 y 3 2 113
Diameter 27 ⇒ radius 7 6. Diameter 43 ⇒ radius 23
x h 2 y k 2 r 2
x h 2 y k 2 r 2
x 3 2 y 1 2 7
x 5 2 y 6 2 12
x 2 y 2 49 8. x 2 y 2 1 9. x 2 2 y 7 2 16
Center: 0, 0
Center: 0, 0
Center: 2, 7
Radius: 7
Radius: 1
Radius: 4
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772

Section 9.1 Circles and Parabolas 773
10.
x 9 2 y 1 2 36 11.
x 1 2 y 2 15 12.
x 2 y 12 2 24
Center:
9, 1
Center:
1, 0
Center:
0, 12
Radius: 6
Radius: 15
Radius: 24 26
13.
1
4 x2 1 4 y2 1 14.
1
9 x2 1 9 y2 1 15.
4
3 x2 4 3 y2 1
x 2 y 2 4
Center: 0, 0
Radius: 2
x 2 y 2 9
Center: 0, 0
Radius: 3
x 2 y 2 3 4
Center: 0, 0
Radius: 3
2
16.
9
2 x2 9 2 y2 1 17.
x 2 y 2 2 9
x 2 2x 1 y 2 6y 9 9 1 9
Center:
x 1 2 y 3 2 1
1, 3
Center: 0, 0
Radius: 2
3
Radius: 1
18.
x 2 10x 25 y 2 6y 9 25 25 9 19.
x 5 2 y 3 2 9
Center: 5, 3
Radius: 3
4x 2 3x 9 4 4y 2 6y 9 41 9 36
Center:
Radius: 1
4x 3 2 2 4y 3 2 4
x 3 2 2 y 3 2 1
3 2 , 3
20.
9 x 2 6x 9 9y 2 4y 4 17 81 36
9x 3 2 9y 2 2 100
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21.
Center:
Radius: 10 3
x 2 y 2 16
Center: 0, 0
Radius: 4
x 3 2 y 2 2 100
9
3, 2
y
x 2 16 y 2 22.
5
3
2
1
−5 −3 −2 −1 1 2 3 5
−2
−3
−5
x
x 2 y 2 81
Center: 0, 0
Radius: 9
y 2 81 x 2
10
8
6
4
2
−10 −6 −4 −2 2 4 6 8 10
−4
−6
−8
−10
y
x

774 Chapter 9 Topics in Analytic Geometry
23.
x 2 4x 4 y 2 4y 4 1 4 4
Center:
Radius: 3
x 2 4x y 2 4y 1 0 24.
x 2 2 y 2 2 9
2, 2
3
2
−7 −6 −5 −3 −2 −1 2 3
−2
−3
−4
−6
−7
y
x
x 2 6x 9 y 2 6y 9 14 9 9
Center:
Radius: 2
x 2 6x y 2 6y 14 0
x 3 2 y 3 2 4
3, 3
1
y
−2 −1 1 2 4 5 6 7 8
−2
−3
−4
−5
−6
−7
−8
−9
x
25.
x 2 14x y 2 8y 40 0
x 2 14x 4 y 2 8y 16 40 49 16
x 7 2 y 4 2 25
6
4
2
y
−2 4 6 8 10
14 16 18
x
Center: 7, 4
Radius: 5
−4
−6
−8
−10
−12
−14
26.
x 2 6x y 2 12y 41 0 27.
x 2 2x y 2 35 0
x 2 6x 9 y 2 12y 36 41 9 36
x 2 2x 1 y 2 35 1
x 3 2 y 6 2 4
x 1 2 y 2 36
28.
Center:
Radius: 2
3, 6
x 2 y 2 10y 25 9 25
Center:
Radius: 4
x 2 y 2 5 2 16
0, 5
9
8
7
6
5
4
3
2
1
−8 −7 −6 −5 −4 −3 −2 −1 1 2
1
−5 −4 −3 −2 −1 1 2 3 4 5
−2
−3
−4
−5
−6
−7
−8
y
y
x
x
Center:
Radius: 6
1, 0
10
8
4
2
−10 −8 −4 −2 2 4 6 8 10
−4
x 2 y 2 10y 9 0 29. y-intercepts:
0 2 2 y 3 2 9
2, 0
−8
−10
4 y 3 2 9
0, 3 ± 5
y 3 2 5
x-intercepts:
x 2 2 0 3 2 9
x 2 2 0
y
y 3 ± 5
x 2
x
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Section 9.1 Circles and Parabolas 775
30. y-intercepts:
0 5 2 y 4 2 25
31. y-intercepts: Let x 0.
0, 4
x-intercepts:
x 5 2 0 4 2 25
x 5 2 16 25
2, 0, 8, 0
y 4 2 0
y 4
x 5 2 9
x 5 ±3
x 8, 2
y 2 6y 27 0
0, 9, 0, 3
x-intercepts: Let y 0.
y 2 6y 9 27 9
y 3 2 36
y 3 ±6
y 9, 3
x 2 2x 27 0
x 2 2x 1 27 1
x 1 2 28
x 1 ±28
x 1 ± 27
1 ± 27, 0
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32. y-intercepts: Let x 0.
33. y-intercepts:
0 6 2 y 3 2 16
y 2 2y 9 0
No solution
No y-intercepts
x-intercepts: Let y 0.
x 2 8x 9 0
x 2 8x 16 9 16
x 4 2 7
x 4 ±7
x 4 ± 7
4 ± 7, 0
No solution
No y-intercepts
x-intercepts:
x 7 2 0 8 2 4
No solution
No x-intercepts
y 8 2 4 49
45
x 7 2 4 64
60
No solution
No y-intercepts
x-intercepts:
x 6 2 0 3 2 16
6 ± 7, 0
34. y-intercepts:
0 7 2 y 8 2 4
35. (a) Radius: 81; Center: 0, 0
y 3 2 16 36
x 6 2 7
20
x 6 ±7
x 6 ± 7
(b) The distance from 60, 45 to 0, 0 is
(c)
x 2 y 2 81 2 6561
60 2 45 2 5625 75 miles.
Yes, you would feel the earthquake.
40
y
(60, 45)
−40 40
−40
x 2 + y 2 = 81 2
x
You were 81 75 6 miles from the outer
boundary.

776 Chapter 9 Topics in Analytic Geometry
36. (a)
Area r 2 1800(b)
r 2 1800
r
1800
r 23.937 feet
R 2 2400
R
2400
27.640 feet
27.640 23.937 3.703 longer radius
37.
y 2 4x 38.
x 2 2y 39.
x 2 8y
Vertex:
0, 0
Vertex:
0, 0
Vertex:
0, 0
Opens to the left since p is
negative.
Matches graph (e).
p 1 2 > 0
Opens upward
Matches graph (b).
Opens downward since p is
negative.
Matches graph (d).
40.
y 2 12x 41.
(y 1) 2 4(x 3) 42.
x 3 2 2y 1
Vertex:
0, 0
Vertex:
3, 1
Vertex:
3, 1
p 3 < 0
Opens to the left
Matches graph (f).
Opens to the right since p is
positive.
Matches graph (a).
p 1 2 < 0
Opens downward
Matches graph (c).
43. Vertex: 0, 0 ⇒ h 0, k 0 44. Point: 2, 6
Graph opens upward.
x ay 2
x 2 4py
Point on graph:
3 2 4p6
9 24p
3
8 p
3, 6
2 a6 2
1 18 a
x 1 18 y2
y 2 18x
45. Vertex: 0, 0 ⇒ h 0, k 0
Focus: 0, 3 2 ⇒ p 3 2
x h 2 4py k
x 2 4 3 2y
x 2 6y
Thus, x 2 4 3 8y ⇒ y 2 3 x2
46. Focus:
y 2 4px 4 5 2x
y 2 10x
⇒ x 2 3 2y.
5 2 , 0 ⇒ p 5 2
47. Vertex: 0, 0 ⇒ h 0, k 0 48. Focus: 0, 1 ⇒ p 1
Focus: 2, 0 ⇒ p 2 x 2 4py 41y
y k 2 4px h
x 2 4y
y 2 42x
y 2 8x
49. Vertex: 0, 0 ⇒ h 0, k 0 50. Directrix: y 3 ⇒ p 3 51. Vertex: 0, 0 ⇒ h 0, k 0
Directrix: y 1 ⇒ p 1 x 2 4py
Directrix: x 2 ⇒ p 2
x h 2 4py k
x 2 12y
y 2 4px
x 0 2 41y 0
y 2 8x
x 2 4y or y 1 4 x2
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Section 9.1 Circles and Parabolas 777
52. Directrix:
y 2 4px
y 2 12x
x 3 ⇒ p 3
53. Vertex: 0, 0 ⇒ h 0, k 0 54. Vertical axis
Horizontal axis and passes through the point 4, 6
y k 2 4px h
Passes through 3, 3
x 2 4py
y 0 2 4px 0
3 2 4p3
y 2 4px
9 12p ⇒ p 3 4
6 2 4p4
x 2 4 3 4y
36 16p ⇒ p 9 4
x 2 3y
y 2 4 9 4x
y 2 9x
55.
y 1 2 x2 56.
y 4x 2 57.
y 2 6x
x 2 2y 4 1 2 y; p 1 2
x 2 1 4 y 4 1 16y, p 1 16
y 2 4 3 2x; p 3 2
Vertex:
Focus:
0, 0
0, 1 2
Vertex: 0, 0
Focus: 0, 16
1
Vertex:
Focus:
0, 0
3
2 , 0
Directrix:
y 1 2
Directrix:
y 1 16
Directrix:
x 3 2
y
y
y
5
4
−2 −1
1 2
x
4
3
3
2
1
–3 –2 2 3
–1
x
−2
−3
−4
–6 –5 –4 –3 –2 –1 1 2
–3
–4
x
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58.
y 2 3x 59.
y 2 4 3 4x; p 3 4
Vertex:
Focus:
Directrix:
4
y
0, 0
3 4 , 0
x 3 4
−2 2 4 6
x
x 2 8y 0 60.
x 2 42y; p 2
Vertex: 0, 0
Focus: 0, 2
Directrix: y 2
y
6
4
2
x
−8 −6 −4
−2
−4
−6
−8
−10
4 6 8
y 2 x
y 2 4 1 4x, p 1 4
Vertex:
Focus:
Directrix:
0, 0
1 4 , 0
x 1 4
–5 –4 –3 –2 –1 1
3
2
–2
–3
y
x

778 Chapter 9 Topics in Analytic Geometry
61.
x 1 2 8y 3 0 62.
x 1 2 42y 3
h 1, k 3, p 2
Vertex: 1, 3
Focus: 1, 5
Directrix: y 1
y
4
2
x
2
−2
−4
−6
−8
−10
−12
x 5 y 4 2 0
y 4 2 x 5 4 1 4x 5
Vertex:
Focus:
5, 4
5 1 4 , 4 19 4 , 4
Directrix: x 21 4
y
1
−2 −1
−1
1 2 3 4 5 6
−2
−3
−4
−5
−6
−7
x
63.
y 2 6y 8x 25 0 64.
y 2 4y 4x 0
y 3 2 42x 2; p 2
y 2 2 4x 1; p 1
Vertex:
2, 3
y
Vertex:
1, 2
y
Focus: 4, 3
Directrix: x 0
–10 –8 –6 –4
2
–2
x
Focus: 0, 2
Directrix: x 2
6
4
–4
–6
–4 2 4
x
–8
–2
65.
x 3 2 2 4y 2 66.
Vertex:
Focus:
⇒ h 3 2 , k 2, p 1
3 2 , 2
3 2 , 2 1 3 2 , 3
x 1 2 2 4y 1 ⇒ p 1
1 2 , 1
1 2 , 1 1 1 2 , 2
Vertex:
Focus:
Directrix:
y 1
6
5
4
y
Directrix: y 0
y
4
3
67.
4y 4 x 1 2
x 1 2 41y 1
h 1, k 1, p 1
Vertex: 1, 1
Focus: 1, 2
Directrix: y 0
−5 −4 −3 −2 −1
6
4
2
y
3
−2
1 2 3
y 1 4x 2 2x 5 68.
–2 2 4
x
x
4x y 2 2y 33 0
y 2 2y 1 4x 33 1
y 1 2 41x 8
Vertex: 8, 1
Focus: 9, 1
Directrix: x 7
−3 −2 −1 1 2
−1
y
6
4
2
2 4 6 10 12
−2
x
x
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−4
−6

Section 9.1 Circles and Parabolas 779
69.
x 2 4x 6y 2 0 70.
x 2 4x 4 6y 2 4 6y 6
x 2 2 6y 1
x 2 2 4 3 2y 1
Vertex: 2, 1
Focus:
2, 1 3 2 2, 1 2
Directrix: y 5 2
y
5
4
3
2
1
x
−6 −4 −3 −1 2
−2
−3
x 2 2x 8y 9 0
Vertex:
Focus:
x 2 2x 1 8y 9 1
1, 1
1,3
Directrix: y 1
x 1 2 8y 1 42y 1
2
−4 −3 −2 −1
−2
−3
−4
−5
−6
−7
−8
y
2 3 4 5 6
x
71.
y 2 x y 0 72.
y 2 y 1 4 x 1 4
y 1 2 2 4 1 4x 1 4
h 1 4 , k 1 2 , p 1 4
Vertex:
Focus:
1 4 , 1 2
0, 1 2
Directrix: x 1 2
To use a graphing
calculator, enter:
y 1 1 2 1 4 x
y 2 1 2 1 4 x
−3
−2
2
1
−1 1
−2
y
x
y 2 4x 4 0
y 2 4x 4 41x 1
Vertex: 1, 0
Focus: 0, 0
Directrix: x 2
y
6
4
2
−4
2 4 6 8
−2
−4
−6
x
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73. Vertex: 3, 1, 74. Vertex:
opens downward
Passes through: 4.5, 4
Passes through: 2, 0, 4, 0
y k 2 4px h
y x 2x 4
y 3 2 4px 5
x 2 6x 8
1 4p4.5 5
x 3 2 1
p 1 2
x 3 2 y 1
y 3 2 2x 5
5, 3 ⇒ h 5, k 3 75. Vertex:
Focus:
y 2 2x 2
2, 0,
opens to the right
3 2 , 0
1
2 p
y 2 4 1 2x 2
76. Vertex: 3, 3 ⇒ h 3, k 3
77. Vertex: 5, 2
Focus: 3, 9 4 ⇒ p 3 4
Focus: 3, 2
x h 2 4py k
Horizontal axis: p 3 5 2
x 3 2 3y 3
y 2 2 42x 5
y 2 2 8x 5

780 Chapter 9 Topics in Analytic Geometry
78. Vertex:
Focus:
1, 2 ⇒ h 1,
k 2
1, 0 ⇒ p 2
x h 2 4py k
x 1 2 42y 2
x 1 2 8y 2
79. Vertex:
Directrix: y 2
Vertical axis
p 4 2 2
0, 4 80. Vertex:
x 0 2 42y 4
x 2 8y 4
Directrix:
2, 1 ⇒ h 2,
k 1
x 1 ⇒ p 3
y k 2 4px h
y 1 2 43x 2
y 1 2 12x 2
81. Focus: 2, 2 82. Focus: 0, 0
Directrix: x 2
Directrix: y 4 ⇒ p 2
Horizontal axis
Vertex: 0, 2
Vertex: 0, 2
x 2 42y 2
p 2 0 2
x 2 8y 2
y 2 2 42x 0
y 2 2 8x
83.
y 2 8x 0x and 3
x y 2 0 84.
y 2 8x and x 2y 3 x 2
y 1 8x
y 2 8x
The point of tangency is
2, 4.
5
−6 6
−3
x 2 12y 0 and x y 3 0
12y x 2
y 1 1 12 x 2
The point of tangency is
6, 3.
y 2 3 x
4
−4 8
−4
85. 4, 8, p 1 x2 2y,
focus:
2 ,
0, 1 2
Following Example 4, we find the y-intercept
0, b.
d 1 1 2 b
d 2 4 0 2 8 1 2 2 17 2
d 1 d 2
⇒
m 8 8
4 0
y 4x 8,
1
2 b 17 2
4
Tangent line
⇒
b 8
Let y 0 ⇒ x 2 ⇒ x-intercept 2, 0.
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Section 9.1 Circles and Parabolas 781
86.
1
4 2 y x2
p 1 2
Focus:
d 1 1 2 b
d 2 3 0 2
9
2 1 2 2 5
1
2 b 5
m
2y x 2 87.
0, 1 2
b 9 2
92 92
0 3
3
y 2x 2 ⇒ x 2 1 2 y 4 1 8 y
Focus:
0, 1 8
Following Example 4, we find the y-intercept
0, b.
d 1 1 8 b
d 2 1 0 2 2 1 8 2 17 8
1
d ⇒ b 2
8 b 17 1 d 2 ⇒
8
m 2 2
1 0 4
y 4x 2
⇒ p 1 8
Let y 0 ⇒ x 1 intercept 1 2 ⇒ x-
2 , 0 .
Tangent line:
y 3x 9 2
⇒ 6x 2y 9 0
x-intercept: 3 2 , 0
88.
y 2x 2 , 2, 8 89.
x 2 1 2 y 4 1 8 y ⇒ p 1 8
Focus:
0, 1 8
R 375x 3 2 x2
R is a maximum of $23,437.50 when
x 125 televisions.
25,000
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d 1 1 8 b
d 2 2 0 2 8 1 8 2 65 8
d 1 d 2 ⇒ 1 8 b 65 8 ⇒ b 8
m 8 8
2 0 8
y 8x 8
Intercept: 1, 0
0 250
0

782 Chapter 9 Topics in Analytic Geometry
90. (a)
x 2 4py
32 2 4p
1
12
1024 1 3 p
3072 p
x 2 43072y
(b)
1
24 x2
12,288
12,288
24
x 2
512 x 2
x 22.6 feet
91. (a)
x 2 4py, p 3 2
x 2 4
3
2 y 6y
or y 2 6x
(b) When x 4,
6y 16
y
6
5
4
8 in.
2 3
(0,
2
1
−4 −3 −2 −1
−1
1 2 3 4
−2
(
x
x2
y
12,288
y 16 6 8 3 .
Depth:
8
3
inches
92. x 2 4py, 6, 4 on parabola
y
93. (a)
y
(b)
x 2 4py
36 4p4
The wire should be inserted
inches from the bottom.
9
4
p 36
16 9 4
10
8
6
2
(6, 4)
−6 −4 −2 2 4 6
−2
x
(−640, 152) (640, 152)
x
640 2 4p152
p 12,800
19
y 19
51,200 x2
(c)
x 0 200 400 500 600
y 0 14.84 59.38 92.77 133.59
94. (a) x 2 4py passes through point 16, 2 5.
96. (a)
256 4p 2 5 ⇒ p 160
x 2 4160y
x 2 640y or y 1
640 x2
(b) 0.1 1
640 x2 ⇒ x 8 feet
V 17,5002 mihr 24,750 mihr 97.
(b) p 4100, h, k 0, 4100
x 0 2 44100y 4100
0x 2 16,400y 4100
95. Vertex:
y 2 4px
Point:
0, 0
1000, 800
800 2 4p1000 ⇒
(a)
y 2 4160x
y 2 640x
p 160
12.5 y 7.125 x 6.25 2
12.5y 89.0625 x 2 12.5x 39.0625
10
0 16
0
y 0.08x 2 x 4
(b) The highest point is at 6.25, 7.125. The
distance is the x-intercept of 15.69 feet.
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Section 9.1 Circles and Parabolas 783
98. (a)
(b)
x 2 1 16 v2 y s 99. The slope of the line joining 3, 4 and the center
3
is 4 3. The slope of the tangent line at 3, 4 is 4.
y 16x2
Thus,
s
v 2
y 4 3
16x2
32 75
4 x 3
2
4y 16 3x 9
1
64 x2 75
3x 4y 25, tangent line.
y 0 1
64 x2 75 ⇒ x 2 7564
⇒ x 69.3 ft
100. The slope of the line joining 5, 12 and the 101. The slope of the line joining 2, 22 and
center is 12 5 . The slope of the tangent line
the center is 222 2. The slope of
5
at 5, 12 is Thus,
the tangent line is 12 22. Thus,
12 .
y 12 5 x 5
12
12y 144 5x 25
5x 12y 169 0, tangent line.
y 22 2 x 2
2
2y 42 2x 22
2x 2y 62, tangent line.
102. The slope of the line joining 25, 2 and the center is 225 15.
The slope of the tangent line is 5. Thus,
y 2 5x 25
y 2 5x 10
5x y 12 0, tangent line.
103. False. The center is 0, 5. 104. True 105. False. A circle is a conic
section.
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106. False. A parabola cannot
intersect its directrix or focus.
109. Answers will vary. See the reflective property of parabolas, page 599.
110. The graph of x 2 y 2 0 is a single point, 0, 0.
5
4
3
2
1
−5 −4 −3 −2 −1 1 2 3 4 5
−2
−3
−4
−5
y
x
107. True 108. False. The directrix y 1 4
is below the x-axis.
The plane intersects the double-napped cone at the vertices of the cones.

784 Chapter 9 Topics in Analytic Geometry
111.
y 3 2 6x 1 112.
For the upper half of the parabola,
y 3 6x 1
y 6x 1 3.
y 1 2 2x 2
For the lower half of the parabola,
y 1 2x 2
y 1 2x 2.
113.
f x 3x 3 4x 2 114.
f x 2x 2 3x
Relative maximum:
0.67, 3.78
Relative minimum:
1.13 at x 0.75
Relative minimum: 0.67, 0.22
115.
f x x 4 2x 2 116.
Relative minimum: 0.79, 0.81
f x x 5 3x 1
Relative minimum: 3.11 at 0.88
Relative maximum: 1.11 at 0.88
Section 9.2
Ellipses
■ An ellipse is the set of all points x, y the sum of whose distances from two distinct fixed points (foci)
is constant.
■ The standard equation of an ellipse with center h, k and major and minor axes of lengths 2a and 2b is
■
(a)
(b)
x h 2
a 2
x h 2
b 2
y k2
b 2 1
y k2
a 2 1
if the major axis is horizontal.
if the major axis is vertical.
c 2 a 2 b 2 where c is the distance from the center to a focus.
■ The eccentricity of an ellipse is e c a .
Vocabulary Check
1. ellipse 2. major axis, center
3. minor axis 4. eccentricity
1.
x 2
4 y 2
9 1 2. x 2
9 y2
4 1 3. x 2
4 y 2
25 1
Center: 0, 0
Center: 0, 0
Center: 0, 0
a 3, b 2
Vertical major axis
Matches graph (b).
a 3, b 2
Horizontal major axis
Matches graph (c).
a 5, b 2
Vertical major axis
Matches graph (d).
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Section 9.2 Ellipses 785
4.
x 2
4 x 2 y2 1 2
5. y 1 2 1 6.
16
Center: 0, 0
Center: 2, 1
a 2, b 1
Horizontal major axis
Matches graph (f).
a 4, b 1
Horizontal major axis
Matches graph (a).
x 2 2 y 22
1
9 4
Center: 2, 2
Horizontal major axis
Matches graph (e).
7.
x 2
y
64 y2
9 1 8. 16 y2
81 1
10
Center: 0, 0
Center: 0, 0
a 8, b 3,
c 64 9 55
Vertices: ±8, 0
Foci: ±55, 0
8
6
4
2
−10 −4 −2 2 4 10
−4
−6
−8
−10
x
x 2
a 9, b 4,
c 81 16 65
Vertices: 0, ±9
Foci: 0, ±65
10
6
4
2
−10−8
−6 −2 2 6 8 10
−4
−6
−10
y
x
e c a 55
8
e c a 65
9
9.
x 4 2 y 12
1 10.
16 25
Center: 4, 1
a 5, b 4, c 3
Vertices:
Foci:
4, 1 ± 5; 4, 6, 4, 4
4, 1 ± 3; 4, 4, 4, 2
x 3 2 y 22
1
12 16
Center: 3, 2
a 4, b 23, c 16 12 2
Foci:
Vertices:
3, 2 ± 2; 3, 0, 3, 4
3, 2 ± 4; 3, 2, 3, 6
e c a 3 5
y
e c a 2 4 1 2
y
6
6
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11.
x 5 2
y 1
94
2 1
Center: 5, 1
Foci:
Vertices:
4
2
−4 −2 2 6 10
−2
−4
−6
−8
a 3 2 , b 1, c 9 4 1 5
2
5
5
2 , 1 , 5
5
2 , 1
5 3 5 3 2 , 1 7 2 , 1 , 2 , 1 13 2 , 1
x
−7 −6 −5 −4 −3 −2 −1
−2
−3
−4
4
3
2
1
y
1
x
4
3
2
1
−7 −4 −3 −2 −1 1
−2
x
e 52
32
5
3

786 Chapter 9 Topics in Analytic Geometry
12.
x 2 2
y 42
14
a 1, b 1 2 , c a2 b 2 3
2
Center: 2, 4
3
Foci:
2
1
2 , 4 ,
3
2
2 , 4
y
–3 –2 –1 1
–1
–2
–3
–4
–5
x
Vertices:
3, 4, 1, 4
Eccentricity: 3
2
13. (a)
x 2 9y 2 36
x 2
36 y2
4 1
(c)
10
8
6
4
y
(b)
a 6, b 2, c 36 4 32 42
Center:
0, 0
Vertices: ±6, 0
Foci: ±42, 0
−10 −8
6 8 10
−4
−6
−8
−10
x
e c a 42
6
22
3
14. (a)
(b)
16x 2 y 2 16
x 2 y2
16 1
a 4, b 1, c 16 1 15
Center:
0, 0
(c)
5
4
1
−5 −4 −3 −2
2 3 4 5
y
x
Vertices: 0, ±4
Foci: 0, ±15
e c a 15
4
15. (a) 9x 2 4y 2 36x 24y 36 0
(c)
(b)
9x 2 4x 4 4y 2 6y 9 36 36 36
a 3, b 2, c 5
Center:
Foci:
2, 3
x 2 2
4
2, 3 ± 5
y 32
9
1
−4
−5
−6 −5 −4 −3 −2 −1
−2
6
4
3
2
y
1 2
x
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Vertices:
2, 6, 2, 0
e 5
3

Section 9.2 Ellipses 787
16. (a) 9x 2 6x 9 4y 2 10y 25 37 81 100 (c)
(b)
a 6, b 4, c 20 25
Center:
Foci:
3, 5
x 3 2
16
3, 5 ± 25
9x 3 2 4y 5 2 144
y 52
36
Vertices: 3, 5 ± 6; 3, 1, 3, 11
e 25 5
6 3
1
2
−2 2 4 8 10 12
−10
−12
y
x
17. (a) 6x 2 2y 2 18x 10y 2 0
(c)
6 x 2 3x
4 9 2 y2 5y 25
4 2 27 2 25
2
4
y
6 x 3 2 2 2 y 5 2 2 24
x 3 2 2
4
y 5 2 2
12
1
−6
−4
2
−2
2
x
(b)
a 23, b 2, c 22
Center:
3 2 , 5 2
Foci:
3 2 , 5 2 ± 22
Vertices:
3 2 , 5 2 ± 23
e 2
3 6
3
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18. (a) x 2 6x 9 4 y 2 5y 25
(c)
4 2 9 25
(b)
a 6, b 3, c 36 9 27 33
Center:
Foci:
3, 5 2
x 3 2 4 y 5 2 2 36
x 3 2
36
3 ± 33, 5 2
y 5 2 2
9
1
−3
6
5
4
3
2
1
y
−1
−2
−3
−4
−6
−7
−8
1 2 3 4 5 6 9 10
x
Vertices:
9, 5 2 , 3, 5 2
e 3
2

788 Chapter 9 Topics in Analytic Geometry
19. (a) 16x 2 25y 2 32x 50y 16 0
(c)
16x 2 2x 1 25y 2 2y 1 16 16 25
(x 1) 2
2516 (y 1)2 1
y
2
1
–2 –1 1 3
x
(b)
a 5 4 , b 1, c 3 4
Center: 1, 1
Foci:
Vertices:
7 4 , 1 , 1 4 , 1
9 4 , 1 , 1 4 , 1
–2
–3
e 3 5
20. (a) 9x 2 25y 2 36x 50y 61 0
(c)
9x 2 4x 4 25y 2 2y 1 61 36 25
9x 2 2 25y 1 2 0
(b) Degenerate ellipse with center 2, 1 as the only point
2
1
y
1 2
x
21. (a) 12x 2 20y 2 12x 40y 37 0
(c)
12 x 2 1 1 4 20y2 2y 1 37 3 20
12 x 1 2 2 20y 1 2 60
y
2
1
−3 −2 −1 1 2 3
−2
x
(b)
a 5, b 3, c 5 3 2
Center:
Vertices:
Foci:
Eccentricity:
1 2 , 1
x 1 2 2
5
1 2 ± 5, 1
1 2 ± 2, 1
c
a 2
5 10
5
y 12
3
1
−3
−4
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Section 9.2 Ellipses 789
22. (a)
36x 2 9y 2 48x 36y 43 0
(b)
a 1, b 1 2 ,
c 1 1 4 3
2
36 x 2 4 3 x 4 9 9y2 4y 4 43 16 36
Center:
2 3 , 2
36 x 2 3 2 9y 2 2 9
Vertices:
2 3 , 2 ± 1 2 3 , 1 , 2 3 , 3
(c)
3
y
x 2 3 2
1
4
y 22
1
1
Foci:
Eccentricity:
2 3 , 2 ± 3
2
c
a 3
2
1
−2 −1
1 2
x
−1
23. Center:
a 4, b 2
0, 0 24. Vertices:
Vertical major axis
x 2
4 y2
16 1
Endpoints of minor axis:
x 2
a 2 y2
b 2 1 0, ±3 2 ⇒ b 3 2
x 2
2 2 y2
32 2 1
±2, 0 ⇒ a 2
x 2
4 4y2
9 1
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25. Center:
0, 0 26. Vertices:
a 3, c 2 ⇒ b 9 4 5
Horizontal major axis
x 2
9 y2
5 1
27. Center:
c 3
a 4 ⇒ b 16 9 7
Horizontal major axis
x 2
16 y2
7 1
Foci:
b 2 a 2 c 2 64 16 48
Center:
y k 2
0, 0 28. Center:
a 2
c 2
0, ±8 ⇒ a 8
0, ±4 ⇒ c 4
0, 0 h, k
0, 0
x h2
b 2 1
y 2
64 x2
48 1
a 6 ⇒ b 36 4 32 42
Horizontal major axis
x 2
36 y2
32 1

790 Chapter 9 Topics in Analytic Geometry
29. Vertices: 0, ±5 ⇒ a 5 30. Vertical major axis
Center: 0, 0
Passes through: 0, 4 and 2, 0
Vertical major axis
a 4, b 2
x h 2
b 2
Point:
4, 2
x 2
b 2 y2
25 1
4 2
b 2 22
25 1
400 21b 2
400
21 b2
x 2
40021 y2
25 1
16
b 1 4 2 25 21
25
21x 2
400 y2
25 1
y k2
a 2 1
x 2
b 2 y2
a 2 1
x 2
4 y2
16 1
31. Center:
a 3, b 1
Vertical major axis
x h 2
b 2
x 2 2
1
2, 3 32. Vertices:
y k2
a 2 1
y 32
9
1
Center:
Endpoints of minor axis:
x h 2
a 2
x 2 2
4
0, 1, 4, 1 ⇒ a 2
2, 1 ⇒ h 2, k 1
y k2
b 2 1
y 12
1
1
2, 0, 2, 2 ⇒ b 1
33. Center:
a 4, b 1 ⇒ c 16 1 15
Horizontal major axis
x 4 2
16
35. Center:
4, 2 34. Center:
y 22
1
0, 4
1
c 4, a 18 ⇒ b 2 a 2 c 2 324 16 308
Vertical major axis
x 2
308 y 42
1
324
c 2, a 3 ⇒ b 2 a 2 c 2 9 4 5
Horizontal major axis
x 2 2
9
2, 0
y2
5 1
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Section 9.2 Ellipses 791
36. Center: 2, 1 ⇒ h 2, k 1 37. Vertices: 3, 1, 3, 9 ⇒ a 4
Vertex: 2, 2 1 ⇒ a 3 Center: 3, 5
2
Minor axis of length 6 ⇒ b 3
Minor axis length: 2 ⇒ b 1
Vertical major axis
x h 2 y k2
1
x h 2 y k2
b 2 a 2 1
b 2 a 2
x 2 2
1
x 2 2
38. Center:
a 3c
Foci:
x h 2
a 2
x 3 2
36
y 12
32 2 1
4y 12
9
y 22
32
1
1, 2, 5, 2 ⇒ c 2, a 6
b 2 a 2 c 2 36 4 32
y k2
b 2 1
1
x 3 2
9
3, 2 h, k 39. Center:
Vertices:
y 52
16
Horizontal major axis
x h 2
a 2
0, 4
1
4, 4, 4, 4 ⇒ a 4
a 2c ⇒ 4 2c ⇒ c 2
2 2 4 2 b 2 ⇒ b 2 12
y k2
b 2 1
x 2 y 42
1
16 12
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40. Vertices:
5, 0, 5, 12 ⇒ a 6
43.
Endpoints of minor axis:
0, 6, 10, 6 ⇒ b 5
Center:
x h 2
b 2
x 5 2
25
x 2 9y 2 10x 36y 52 0
x 2 10x 25 9 y 2 4y 4 52 25 36
a 3, b 1, c 9 1 22
e c a 22
3
5, 6 ⇒ h 5, k 6
y k2
a 2 1
y 62
36
1
x 5 2 9 y 2 2 9
x 5 2
9
y 22
1
41.
1
x 2
4 y2
9 1 42. x 2
25 y2
36 1
a 3, b 2,
a 6, b 5,
c 9 4 5
c 36 25 11
e c a 5
e c 3
a 11
6

792 Chapter 9 Topics in Analytic Geometry
44.
4x 2 3y 2 8x 18y 19 0
4 x 2 2x 1 3 y 2 6y 9 19 4 27
4 x 1 2 3 y 3 2 12
x 1 2
3
y 32
4
a 2, b 3, c 4 3 1
1
e c a 1 2
45. Vertices:
Eccentricity:
b 2 a 2 c 2 25 16 9
Center:
0, 0
Horizontal major axis
x 2
25 y2
9 1
±5, 0 ⇒ a 5
4
5 c a ⇒ c 4 5 a 4
46. Vertices:
Eccentricity:
b 2 a 2 c 2 64 16 48
x 2
b 2 y2
a 2 1
x 2
48 y2
64 1
0, ±8 ⇒ a 8, h 0, k 0
e 1 2 c a
1
2 c 8
c 4
47. (a)
48. (a)
80
60
20
(0, 40)
(−50, 0) (50, 0)
−40 −20 20 40
−20
y
−20 −12−8 −4 4 8 12 16 20
−8
−12
−16
−20
x 2
20
16
y
(0, 12)
8
(−16, 0)
4
(16, 0)
x
x
(b) Vertices:
(b)
Height at center:
40 ⇒ b 40
Horizontal major axis
x 2
a 2 y2
b 2 1
±50, 0 ⇒ a 50
x 2
2500 y2
1600 1, y ≥ 0
a 16, b 12
x 2
256 y2
144 1, y ≥ 0
49. Let be the equation of the ellipse. Then b 2 and
a y2
2 b 1 2
a 3 ⇒ c 2 a 2 b 2 9 4 5. Thus, the tacks are placed
at ±5, 0. The string has a length of 2a 6 feet.
(c) For
x 45, 45 2
2500 y2
1600 1.
y 2 1600 1
2500
452
y 2 304
y 17.44
The height five feet from
the edge of the tunnel is
approximately 17.44 feet.
(c) When x 10,
y 2 144 1 102
256
y 9.4 > 9.
Hence, the truck will be able
to drive through without
crossing the center line.
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Section 9.2 Ellipses 793
50.
40
x 2
972 2 y2
y
−20 20 40
−40
23 2 1 x
x2
or 23 y2
2 972 1 2
a 97 2 , b 23, c 97 2 2 23 2 4.7
51.
Area of ellipse 2area of circle
ab 2r 2
a10 210 2
a10 200
a 20
Length of major axis: 2a 220 40 units
Distance between foci:
24.7 85.4 feet
52. Center:
2a 35.88 ⇒ a 17.94 ⇒ a 2 321.84
e c a ⇒ 0.97
c 2 a 2 b 2 ⇒ b 2 a 2 c 2 19.02
Ellipse:
0, 0, e 0.97 53.
c
17.94 ⇒ c 17.4018
x 2
321.84 y 2
19.02 1
a c 4.08
a c 0.34
2a 4.42 ⇒ a 2.21 ⇒ c 1.87
b 2 a 2 c 2 ⇒ b 2 1.3872
x 2
4.8841 y 2
1.3872 1
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54.
56.
a c 947 6378 7325 55. For we have c 2 a 2 b 2 .
a y2
2 b 1, 2
a c 228 6378 6606
When x c,
y
2a 13,931
a − c
c 2
a 6965.5
b
a y2
2 b 1 ⇒ 2 y2 b 2 1 a2 b 2
a 2
c 7325 6965.5
359.5
e c a 0.0516
2b 2
a 1
−a
a + c
Additional points: 3, ± 1 2 , 3, ±1 2
−b
−2
c
a
x
x 2
⇒ y 2 b4
a 2
⇒ 2y 2b2
a .
x 2
y
4 y2
1 1 57. x 2
9 y2
16 1
2
a 2, b 1, c 3
1
1
( − 3,
2
)
( 3,
2
) a 4, b 3, c 7
Points on the ellipse:
x Points on the ellipse:
−1
1
±2, 0, 0, ±1
±3, 0, 0, ±4
1
1
( − 3, −
( −
2
)
3,
2
)
Length of latus recta:
Length of latus recta:
2b 2
a 232
4
9 2
9
− , 7
4
−2
9 9
− , ,
4
− 7 4
− 7
Additional points: ± 9 4 , 7 , ±9 4 , 7
(
)
2
9
, 7
4
−4 −2
2 4
( ) ( )
y
(
)
x

794 Chapter 9 Topics in Analytic Geometry
y
58. 9x 2 4y 2 36
y
59. 5x 2 3y 2 15
x 2
4 y2
4
4
( − , 5 ( , 5
9 1 3
)
3
)
x 2
2
3 y 2
4
5 1 3 5
3 5
( − , 2 ) ( , 2
5
x
Points on the ellipse:
−3 −1 1 3
a 5, b 3,
−4 −2 2 4
3 5
3 5
±2, 0, 0, ±3
( − , − 2
−2
5
) (
5
4
4
c 2
( − , − 5 ( , − 5
3
)
3
)
−4
Length of latus recta:
2b 2
a 2 Points on the ellipse:
22
8 ±3, 0, 0, ±5
3 3
2b 2
Length of latus recta:
a 2 3
5 65
5
Additional points: ± 4 3 , 5 , ±4 3 , 5
5
)
, − 2
Additional points: ± 35
5 , 2 , ±35 5 , 2
x
)
60. Answers will vary. 61. True. If e 1 then the ellipse is elongated, not
circular.
62. True. The ellipse is inside the circle. 63. (a) The length of the string is 2a.
(b) The path is an ellipse because the sum of the
distances from the two thumbtacks is always
the length of the string, that is, it is constant.
64. (a)
(b)
a b 20 ⇒ b 20 a
A ab a20 a
a 2 20a 264 0
a 14 or a 6 by the Quadratic Formula
b 64 ORb 14
Since a > b, we choose
x 2
14 2 y 2
x 2
196 y 2
36 1
264 a20 a
6 2 1 a 14 and b 6.
65. Center: 6, 2 66.
Foci: 2, 2, 10, 2 ⇒ c 4
a c a c 2a 36 ⇒ a 18
b 2 a 2 b 2 ⇒ b 18 2 16 308
Horizontal major axis
x 6 2
324
y 22
308
1
(c)
(d)
a
A
360
8 9 10 11 12 13
301.6 311.0 314.2 311.0 301.6 285.9
0 24
0
The area is maximum when a b 10 and
it is a circle.
x 2
a 2 y2
b 2 1
The sum of the distances
from any point on the
ellipse to the two foci is
constant. Using the vertex
a, 0, you have
a c a c 2a.
From the figure,
2b 2 c 2 2a ⇒ a 2 b 2 c 2 .
−a
−c
b
b
−b
y
c
b 2 + c 2
c
a
x
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Section 9.3 Hyperbolas 795
67. Arithmetic: d 11 68. Geometric: r 1 2 69. Geometric: r 2 70. Arithmetic: d 1
71. 6
3 n 1093 72. 6
3 n 547 73. 10
n0
n0
n1
4 3 4 n1 15.099 74. 10
n0
5 4 3 n 340.155
Section 9.3
Hyperbolas
■ A hyperbola is the set of all points x, y the difference of whose distances from two distinct fixed points
(foci) is constant.
■ The standard equation of a hyperbola with center h, k and transverse and conjugate axes of lengths
2a and 2b is:
■
■
(a)
(b)
x h 2
a 2
y k 2
a 2
y k2
b 2 1
x h2
b 2 1
if the transverse axis is horizontal.
if the transverse axis is vertical.
c 2 a 2 b 2 where c is the distance from the center to a focus.
The asymptotes of a hyperbola are:
(a)
(b)
y k ± b x h
a
y k ± a x h
b
if the transverse axis is horizontal.
the transverse axis is vertical.
■
■
The eccentricity of a hyperbola is e c a .
To classify a nondegenerate conic from its general equation Ax 2 Cy 2 Dx Ey F 0:
(a) If A C (A 0, C 0), then it is a circle.
(b) If AC 0 (A 0 or C 0, but not both), then it is a parabola.
(c) If AC > 0, then it is an ellipse.
(d) If AC < 0, then it is a hyperbola.
© Houghton Mifflin Company. All rights reserved.
Vocabulary Check
1. hyperbola 2. branches 3. transverse axis, center
4. asymptotes 5. Ax 2 Cy 2 Dx Ey F 0
1. Center: 0, 0 2. Center: 0, 0
a 3, b 5, c 34
a 5, b 3
Vertical transverse axis
Matches graph (b).
Vertical transverse axis
Matches graph (c).
3. Center: 1, 0 4. Center: 1, 2
a 4, b 2
a 4, b 3
Horizontal transverse axis
Matches graph (a).
Horizontal transverse axis
Matches graph (d).

796 Chapter 9 Topics in Analytic Geometry
5.
y
x 2 y 2 1 6.
a 1, b 1, c 2
Center: 0, 0
Vertices: ±1, 0
Foci: ±2, 0
Asymptotes: y ±x
2
1
–2 2
–1
–2
x
x 2
9 y2
25 1
Center: 0, 0
a 3, b 5,
c 3 2 5 2 34
Vertices: ±3, 0
Foci: ±34, 0
10
8
6
4
−8 −6 −4 2 4 6 8 10
−6
−8
−10
y
x
Asymptotes: y ± b a x ±5 3 x
7.
y 2
1 x2
a 1, b 2, c 5
3
2
y
y 2
9 x2
1 1
a 3, b 1,
y
6
4 1 8. Center:
Center:
Vertices:
Foci:
0, 0
0, ±1
0, ±5
Asymptotes: y ± 1 2 x
–3 –2 2 3
–2
–3
x
c 3 2 1 2 10
0, 0
Vertices: 0, ±3
Foci: 0, ±10
Asymptotes: y ±3x
–6 –4 –2 2 4 6
–6
x
9.
y 2
25 x2
81 1 10. x 2
36 y2
4 1
a 5, b 9, c a 2 b 2 106
a 6, b 2,
y
Center: 0, 0
c 36 4 210
15
Vertices: 0, ±5
12
Center: 0, 0
9
Foci: 0, ±106
Vertices: ±6, 0
3
x
Asymptotes:
−9 −6
−3
6 9 12 15
Foci: ±210, 0
y
12
8
4
–12 12
–4
–8
–12
x
11.
y ± a b x ±5 9 x
x 1 2
4
a 2, b 1, c 5
Center:
Vertices:
1, 2, 3, 2
Foci:
y 22
1
1, 2
1 ± 5, 2
Asymptotes: y 2 ± 1 x 1
2
3
2
1
–4
–5
−9
−12
−15
y
1 12.
1 2 3
x
Asymptotes: y ± 1 3 x
x 3 2 y 22
1
144 25
Center: 3, 2
a 12, b 5, c 13
Vertices: 15, 2, 9, 2
Foci: 16, 2, 10, 2
Asymptotes:
y 2 ± 5 x 3
12
15
10
5
y
−5 5
−5
−10
−15
−20
x
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Section 9.3 Hyperbolas 797
13.
y 5 2
19
a 1 3 , b 1 2 , c 1 9 1 4 13
6
Center:
Vertices:
Foci:
1, 5
Asymptotes:
x 12
14
1, 5 ± 1 3 : 1, 16
3 , 1, 14 3
13
1, 5 ±
6
1 14.
y k ± a x h
b
y 5 ± 2 x 1
3
y 1 2 x 32
1
14 116
Center: 3, 1
a 1 2 , b 1 4 , c 1 4 1 16 5
4
Vertices:
Foci:
Asymptotes:
3, 1 2 , 3, 3 2
5
3, 1 ±
4
y 1 ± 12 x 3 1 ± 2x 3
14
3
y
y
2
−2 −1 1 2 3 4
−1
−2
x
–3 –1
1
x
−3
–1
−5
15. (a)
4x 2 9y 2 36
(b) Center:
x 2
9 y2
4 1
0, 0
a 3, b 2, c 9 4 13
Vertices: ±3, 0
Foci: ±13, 0
16. (a)
25x 2 4y 2 100
(b) Center:
x 2
4 y2
25 1
0, 0
a 2, b 5, c 4 25 29
Vertices: ±2, 0
Foci: ±29, 0
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(c)
Asymptotes:
5
4
3
2
1
−5 −4 −2
2 4 5
−2
−3
−4
−5
y
y ± b a x ±2 3 x
x
(c)
Asymptotes: y ± b a x ±5 2 x
y
8
6
4
x
−8 −6 −4
4 6 8
−6
−8

798 Chapter 9 Topics in Analytic Geometry
17. (a)
2x 2 3y 2 6
(c) To use a graphing calculator, solve first for y.
x 2
3 y2
2 1
y 2 2x2 6
3
(b)
a 3, b 2, c 5
Center:
0, 0
Vertices: ±3, 0
Foci: ±5, 0
Asymptotes: y ±
2
3 x
± 6
3 x
y 1
2x 2 6
3
y 2
2x 2 6
3
y 3
2
3 x
y 4
2
3 x
Hyperbola
Asymptotes
4
3
2
1
−4 −3
3 4
−2
−3
−4
y
x
18. (a)
6y 2 3x 2 18
(c)
y
(b)
y 2
3 x 2
6 1
a 3, b 6, c 3
Center:
Vertices:
0, 0
0, ±3
4
3
1
−4 −3 −2
−1
−3
−4
2 3 4
x
Foci:
0, ±3
Asymptotes: y ± 3
6 x ±2 2 x
19. (a)
9x 2 y 2 36x 6y 18 0
(c)
y
9x 2 4x 4 y 2 6y 9 18 36 9
2
x 2 2
1
y 32
9
1
–6 –4 –2 2 4 6 8
–4
x
(b)
20. (a)
a 1, b 3, c 10
Center: 2, 3
Vertices: 1, 3, 3, 3
Foci: 2 ± 10, 3
Asymptotes: y 3 ± 3x 2
x 2 9y 2 36y 72 0
x 2 9 y 2 4y 4 72 36
x 2 9y 2 2 36
x 2 y 22
1
36 4
(b)
a 6, b 2,
Center:
0, 2
Vertices: ±6, 2
Foci: ±210, 2
–6
–8
c 36 4 210
(c)
y
12
8
4
–8 –4 4 8
–4
–8
–12
x
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Asymptotes: y 2 ± 1 3 x

Section 9.3 Hyperbolas 799
21. (a)
x 2 9y 2 2x 54y 80 0
(c)
y
x 2 2x 1 9y 2 6y 9 80 1 81
x 1 2 9y 3 2 0
y 3 ± 1 3x 1
4
2
–4 –2 2
–2
x
(b) Degenerate hyperbola is two lines intersecting at
1, 3.
–4
–6
22. (a)
16y 2 x 2 2x 64y 63 0
(c)
y
16y 2 4y 4 x 2 2x 1 63 64 1
–1 1 2 3
x
16y 2 2 x 1 0
–1
y 2 ± 1 4x 1
(b) Degenerate hyperbola is two intersecting lines at
1, 2.
–2
–3
–4
23. (a)
(b)
Center:
Vertices:
Foci:
9y 2 x 2 2x 54y 62 0
9y 2 6y 9 x 2 2x 1 62 1 81
1, 3
y 3 2
2
1, 3 ± 2
1, 3 ± 25
x 12
18
a 2, b 32, c 25
1
Asymptotes:
y 3 ± 1 x 1
3
(c) To use a graphing calculator, solve for y first.
y
9y 3 2 18 x 1 2
4
2
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y 3 ±
18 x 1 2
y 1 3 1 18 x 12
3
y 2 3 1 18 x 12
3
y 3 3 1 x 1
3
y 4 3 1 x 1
3
9
Hyperbola
Asymptotes
−6
−8
−10
2
x

800 Chapter 9 Topics in Analytic Geometry
24. (a)
9x 2 y 2 54x 10y 55 0
(c)
y
9x 2 6x 9 y 2 10y 25 55 81 25
14
(b)
x 3 2
19
a 1 10
, b 1, c
3 3
y 52
1
1
10
8
6
4
2
−7 −6 −4 −3 −2 1
x
Center:
3, 5
Vertices:
3 ± 1 3 , 5
Foci:
10
3 ±
3 , 5
Asymptotes: y 5 ± 3x 3
25. Vertices:
Foci:
Center:
y k 2
a 2
0, ±2 ⇒ a 2 26. Vertices:
0, ±4 ⇒ c 4
b 2 c 2 a 2 16 4 12
0, 0 h, k
x h2
b 2 1
y 2
4 x2
12 1
Foci:
x 2
a 2 y2
b 2 1
x 2
9 y2
27 1
±3, 0 ⇒ a 3
±6, 0 ⇒ c 6
b 2 c 2 a 2 36 9 27
27. Vertices: ±1, 0 ⇒ a 1
Asymptotes:
y ±5x ⇒ b a 5
Center:
x 2
1 y2
25 1
⇒ b 5
0, 0
28. Vertices:
0, ±3 ⇒ a 3 29. Foci:
0, ±8 ⇒ c 8
Asymptotes:
y ±3x ⇒ a b 3, b 1
Asymptotes:
y ±4x ⇒ a b 4 ⇒ a 4b
Center:
y k 2
a 2
0, 0 h, k
x h2
b 2 1
y 2
9 x2 1
Center:
c 2 a 2 b 2 ⇒ 64 16b 2 b 2
64
17 b2 ⇒ a 2 1024
17
y k 2
a 2
0, 0 h, k
x h2
b 2 1
y 2
102417
x2
6417 1
17y 2
1024 17x2
64 1
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Section 9.3 Hyperbolas 801
30. Foci:
Asymptotes:
±10, 0 ⇒ c 10
c 2 a 2 b 2 ⇒ 100 3m 2 4m 2
a 42 8, b 32 6
x 2
a y2
2 b 1 2
x 2
64 y2
36 1
y ± 3 4 x ⇒ b a 3m
4m
100 25m 2
2 m
31. Vertices:
Foci:
b 2 c 2 a 2 16 4 12
Center:
x h 2
a 2
2, 0, 6, 0 ⇒ a 2
0, 0, 8, 0 ⇒ c 4
4, 0 h, k
x 4 2
4
y k2
b 2 1
y 2
12 1
32. Vertices: 2, 3, 2, 3 ⇒ a 3
33. Vertices: 4, 1, 4, 9 ⇒ a 4
Center: 2, 0
Foci: 4, 0, 4, 10 ⇒ c 5
Foci:
b 2 c 2 a 2 25 9 16
y k 2
a 2
2, 5, 2, 5 ⇒ c 5
y 2
9
x h2
b 2 1
x 22
16
1
b 2 c 2 a 2 25 16 9
Center:
y k 2
a 2
y 5 2
16
4, 5 h, k
x h2
b 2 1
x 42
9
1
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34. Vertices:
Center:
Foci:
b 2 c 2 a 2 9 4 5
x h 2
a 2
3, 1, 3, 1 ⇒ c 3
x 2
4
2, 1, 2, 1 ⇒ a 2
0, 1
y k2
b 2 1
y 12
5
1
35. Vertices:
Solution point:
Center:
y k 2
a 2
y 2
9
y 2
9
2, 3, 2, 3 ⇒ a 3
0, 5
2, 0 h, k
x h2
b 2 1
x 22
b 2
x 22
94
b 2
1 ⇒
922
25 9 36
16 9 4
1
9x 22
y 2 9

802 Chapter 9 Topics in Analytic Geometry
36. Center:
x 2
4
Solution point:
x 2
4
0, 1, a 2 37. Vertices:
y 12
b 2 1
25
4 9 b 2 1
y 12
127
5, 4
9
b 21
2 4
b 2 36
21 12 7
1
Center:
y 2 2
4
Passes through 5, 1
1 2 2
5 4 b 1 2
y 2 2
4
0, 4, 0, 0
0, 2, a 2
x 2
b 2 1
9
4 1 5 b 2
b 2 4 ⇒ b 2
x2
4 1
38. Center:
y 2
4
Solution point:
y 2
4
1, 0, a 2 39. Vertices:
x 12
b 2 1
5
4 1 b 2 1
x 12
4
0, 5
1
b 2 1 4 ⇒ b 2
1
1, 2, 3, 2 ⇒ a 1
Center:
Asymptotes:
x 2 2
1
2, 2
y x, y 4 x
b
a 1 ⇒ b 1
y 22
1
1
40. Center:
Asymptotes:
y 3 2
9
3, 3, a 3
y x 6, y x
1 a b 3 b ⇒ b 3
x 32
9
1
41. Vertices:
0, 2, 6, 2 ⇒ a 3 42. Vertices:
(3, 0, 3, 4 ⇒ a 2
Asymptotes:
y 2 3 x, y 4 2 3 x
Asymptotes:
y 2 3 x, y 4 2 3 x
b
a 2 3 ⇒ b 2
Center:
x h 2
a 2
x 3 2
9
3, 2 h, k
y k2
b 2 1
y 22
4
1
a
b 2 3 ⇒ b 3
Center:
y k 2
a 2
y 2 2
4
3, 2 h, k
x h2
b 2 1
x 32
9
1
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Section 9.3 Hyperbolas 803
43. F 1 : Friend’s location 10,560, 0
F 2 : Your location 10,560, 0
Px, y:
x 2
a 2 y2
b 2 1
Location of lightning strike
110018 19,800
c 10,560, a 19,800
2
b 2 c 2 a 2 13,503,600
x 2
98,010,000 y 2
13,503,600
9900 ⇒ a 2 98,010,000
y
10,000
P
Friend
F1 F2
You
−20,000 20,000
(−10,560, 0) (10,560, 0)
−10,000
x
y
44. The explosion occurred on the vertical line through
(3300, 1100)
3300, 1100 and 3300, 0.
4000
d 2 d 1 41100 4400
Hence,
2a 4400
a 2200
c 3300
b 2 c 2 a 2 .
The explosion occurred on the hyperbola
Letting x 3300,
y 2 b 2 x2
a 2 1 33002 2200 2
3300 2
2200 2 1 ⇒ y 2750.
3300, 2750
x 2
a 2 y2
b 2 1.
( −3300, 0)
−4000
3000
2000
1000
d 2
−4000
a
(3300, 0)
d 1
x
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x
45. (a)
2
a y2
2 b 1 2
a 1; 2, 9 is on the curve, so
4
1 81
b 1 ⇒ 81
2 b 3 2
⇒ b 2 81
3 ⇒ b 33.
x 2
1 y2
1, 9 ≤ y ≤ 9
27
1
2
(b) Because each unit is 2 foot, 4 inches is 3 of a unit.
The base is 9 units from the origin, so
y 9 2 3 81 3 .
When y 25 3 ,
x 2 1 2532
27
⇒ x 1.88998.
So the width is 2x 3.779956 units, or
22.68 inches, or 1.88998 feet.

804 Chapter 9 Topics in Analytic Geometry
46. Foci: ±150, 0 ⇒ c 150
Center: 0, 0
y
150
75
(x, 75)
(−150, 0)
−150
−75
d 2 d 1
(150, 0)
x
75 150
(a)
d 2 d 1 186,0000.001
186 ⇒ 2a 186 ⇒ a 93
b 2 c 2 a 2 150 2 93 2 13,851
x 2
93 2 y2
13,851 1
x 110.3 miles
(b) 150 93 57 miles
x 2 93 2 1 13,851 752 12,161.43
(c) Bay to Station 1: 30 miles
Bay to Station 2: 270 miles
270 30
186,000
(d) In this case,
d 2 d 1 186,0000.00129 239.94 ⇒ a 120
and b 2 c 2 a 2 8100. The hyperbola is
x 2
120 2 y2
90 2 1.
0.00129 second
For y 60, x 2 20,800 and x 144.2.
Position: 144.2, 60
47. Center:
Focus:
0, 0
24, 0
b 2 c 2 a 2 24 2 a 2 576 a 2
x 2
a 2 y 2
576 a 2 1
24 2
a 2 242
576 a 2 1
576
a 2 576
576 a 2 1
576576 a 2 576a 2 a 2 576 a 2
a 4 1728a 2 331,776 0
a ±38.83 or a ±14.83
Since a < c and c 24, we choose a 14.83. The vertex is approximate at 14.83, 0.
[Note: By the Quadratic Formula, the exact value of a is a 125 1. ]
48.
x 2
25 y2
16 1
a 5, b 4, c 25 16 41
9x 2 4y 2 18x 16y 119 0
A 9, C 4
The camera is 5 41 units from the mirror.
AC 36 > 0, Ellipse
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Section 9.3 Hyperbolas 805
50.
x 2 y 2 4x 6y 23 0 51.
A C 1,
Circle
16x 2 9y 2 32x 54y 209 0
A 16, C 9
AC 169 < 0, Hyperbola
52.
x 2 4x 8y 20 0 53.
A 1, C 0
AC 0, Parabola
y 2 12x 4y 28 0
C 1, A 0
AC 0, Parabola
54.
4x 2 25y 2 16x 250y 541 0 55.
A 4, C 25
AC 100 > 0, Ellipse
x 2 y 2 2x 6y 0
A C 1,
Circle
56.
y 2 x 2 2x 6y 8 0 57.
A 1, C 1
AC < 0, Hyperbola
x 2 6x 2y 7 0
A 1, C 0, D 6,
AC 0 ⇒ Parabola
E 2, F 7
58.
9x 2 4y 2 90x 8y 228 0 59. True. e c a a2 b 2
A 9, C 4
AC 94 36 > 0 ⇒ Ellipse
a
60. False. b 0 because it is in the denominator.
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61. False. For example,
x 2 y 2 2x 2y 0
x 1 2 y 1 2 0
is the graph of two intersecting lines.
62. True. The asymptotes are
y ± b a x.
63. Let x, y be such that the difference of the distances from c, 0 and c, 0
is 2a (again only deriving one of the forms).
2a x c 2 y 2 x c 2 y 2
4a 2 4ax c 2 y 2 x c 2 y 2 x c 2 y 2
4ax c 2 y 2 4cx 4a 2
ax c 2 y 2 cx a 2
a 2 x 2 2cx c 2 y 2 c 2 x 2 2a 2 cx a 4
a 2 c 2 a 2 c 2 a 2 x 2 a 2 y 2
If they intersect at right angles, then
b
a 1
ba a b ⇒ a b.
2a
x c 2 y 2 x c y 2
Let b2 c 2 a 2 .
Then a 2 b 2 b 2 x 2 a 2 y 2 ⇒ 1 x2
a 2 y2
b 2.

806 Chapter 9 Topics in Analytic Geometry
64. Answers will vary. See Example 3. 65.
66. Center:
Horizontal transverse axis
Foci at 2, 2 and 10, 2 ⇒ c 4.
c a c a 6 ⇒ a 3
b 2 c 2 a 2 16 9 7
x 6 2
9
6, 2
y 22
7
1
d 2 d 1
At the point a, 0,
d 2 d 1 a c c a 2a.
constant by definition of hyperbola
67. At the point a, 0, the difference of the distances to the foci ±c, 0 is c a c a 2a.
Let x, y be a point on the hyperbola.
2a x c 2 y 2 x c 2 y 2
2a x c 2 y 2 x c 2 y 2
4a 2 4ax c 2 y 2 x c 2 y 2 x c 2 y 2
4ax c 2 y 2 4cx 4a 2
ax c 2 y 2 cx a 2
a 2 x 2 2cx c 2 y 2 c 2 x 2 2a 2 cx a 4
a 2 c 2 a 2 c 2 a 2 x 2 a 2 y 2
Thus, c 2 a 2 b 2 , as desired.
1 x2
a
y2
2 c 2 a 2
68. If A C 0, then by completing the square you obtain a circle.
If A 0 and C 0, then Cy 2 Dx Ey F 0 is a parabola (complete the square).
Same for A 0 and C 0.
If AC > 0, then both A and C are positive (or both negative). By completing the square
you obtain an ellipse.
If AC < 0, then A and C have opposite signs. You obtain a hyperbola.
69. x 3 3x 2 6 2x 4x 2 x 3 x 2 2x 6 70.
71.
2 1
1
x 3 3x 4
x 2
0
2
2
3
4
1
4
2
2
x 2 2x 1 2
x 2
72.
3x 1 2x 4 3x 2 12x 1 2 x 2
3x 2 23 2 x 2
x y 3 2 x y 2 6x y 9
x 2 2xy y 2 6x 6y 9
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Section 9.4 Rotation and Systems of Quadratic Equations 807
73. x 3 16x xx 2 16 xx 4x 4 74. x 2 14x 49 x 7 2
75.
2x 3 24x 2 72x 2xx 2 12x 36 76.
2xx 6 2
6x 3 11x 2 10x x6x 2 11x 10
x3x 22x 5
77.
16x 3 54 28x 3 27 78.
22x 34x 2 6x 9
4 x 4x 2 x 3 4 x x 2 4 x
4 xx 2 1
4 xx ix i
Section 9.4
Rotation and Systems of Quadratic Equations
■ The general second-degree equation Ax 2 Bxy Cy 2 Dx Ey F 0 can be rewritten
Ax 2 Cy 2 Dx Ey F 0 by rotating the coordinate axes through the angle
where cot 2 A CB.
■ x x cos ysin
y x sin y cos
■ The graph of the nondegenerate equation Ax 2 Bxy Cy 2 Dx Ey F 0 is:
(a) An ellipse or circle if B 2 4AC < 0.
(b) A parabola if B 2 4AC 0.
(c) A hyperbola if B 2 4AC > 0.
Vocabulary Check
1. rotation, axes 2. invariant under rotation 3. discriminant
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90;
1. Point:
x x cos y sin
0 x cos 90 y sin 90
0 y
0, 3
Thus, x, y 3, 0.
45;
2. Point:
3, 3
x x cos y sin
3 x cos 45 y sin 45
3 2
2 x 2
2 y
y x sin y cos
3 x sin 90 y cos 90
3 x
Adding, 6 2x ⇒ x 6 2 32.
y x sin y cos
3 x sin 45 y cos 45
3 2
2 x 2
2 y
2y 0 ⇒ y 0.
Subtracting,
Thus, x, y 32, 0.

808 Chapter 9 Topics in Analytic Geometry
3.
xy 1 0
A 0, B 1, C 0
y
4
y′ x′
cot 2 A C
B
x x cos
4 y sin
4
y x sin
4 y cos
4
0 ⇒ 2
2 ⇒
x 2 2
2 y 2
x 2 2
2 y 2
4
x y
2
x y
2
−4 −3 −2 4
−2
−3
−4
x
xy 1 0
x y
2 x y
2 1 0
y 2
2
x 2
2
1,
Hyperbola
4.
xy 2 0, A 0, B 1, C 0
cot 2 A C
B
x x cos
4 y 2
sin
4
x 2
y x sin
4 y 2
cos
4
x 2
0 ⇒ 2
2 ⇒
10
y′
8
6
4 4
y
2
2 x y
2
y
2
2 x y
2
−8
−10
y
x′
4 6 8 10
x
xy 2 0
x y
2 x y
2 2 0
x 2 y
2
2
2
5.
x 2
4 y 2
4
x 2 4xy y 2 1 0
x 2 2
2 y 2
1, Hyperbola
4
−8 −6 −4
4 6 8
A 1, B 4, C 1
y x sin cot 2 A C 0 ⇒ 2
B
2 ⇒
−6
4 y cos
4
−8
y
y′
8
x′
6
x
x x cos
4 y sin
4
4
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2
2 x y

Section 9.4 Rotation and Systems of Quadratic Equations 809
5. —CONTINUED—
x 2 4xy y 2 1 0
2
2 x y
2
4 2
2 x y 2
2 x y
2
2 x y
2
1 0
1
2 x 2 xy 1 2 y 2 2x 2 y 2 1 2 x 2 xy 1 2 y 2 1 0
x 2 3y 2 1
x 2 y2
, Hyperbola
13 1
6.
xy x 2y 3 0
y
A 0, B 1, C 0
y′
8
6
x′
cot 2 A C
B
x x cos
4 y sin
4
x 2 2
2 y 2
0 ⇒ 2
2 ⇒
4
y x sin
4 y cos
4
2 2
x 2 y 2
−8 −6 −4 4 6
4
−4
−6
−8
x
x y
2
x y
2
xy x 2y 3 0
x y
2 x y
2 x y
2 2 x y
2 3 0
x 2
2
y 2
2
x
2 y
2 2x
2 2y
2 3 0
x 2 2x
2
2 2 y 2 32y
32
2 2 6 2
2 2
32
2 2
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x 2
2 2 y 32
2 2 10
y 32
2 2
10
x 2
2 2
10
1 , Hyperbola

810 Chapter 9 Topics in Analytic Geometry
7.
xy 2y 4x 0
y
A 0, B 1, C 0
cot 2 A C
B
0 ⇒ 2
2 ⇒
4
y′
8
6
4
x′
x x cos
4 y sin
4
x 2 2
2 y 2
y x sin
4 y cos
4
x 2 2
2 y 2
−4 2 4 6 8
−4
x
x y
2
x y
2
x y
2 x y
2 2 x y
2 4 x y
2 0
x 2
2
y 2
2
2x 2y 22x 22y 0
x 2 62x 32 2 y 2 22y 2 2 0 32 2 2 2
x 32 2 y 2 2 16
x 32 2
16
xy 2y 4x 0
y 2 2
16
1, Hyperbola
8.
2x 2 3xy 2y 2 10 0
y
x′
A 2, B 3, C 2
4
cot 2
C
B
4 3 ⇒
71.57
2
y′
−4 −2 4
x
cos 2 4 5
sin
1 cos 2
2
1 45
3
2 10
cos
1 cos 2
2
1 45
1
2 10
x x cos y sin y x sin y cos
x10 1 y 10 3
x 10 3 y 10
1
x 3y
10
—CONTINUED—
3x y
10
−4
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Section 9.4 Rotation and Systems of Quadratic Equations 811
8. —CONTINUED—
x 2
5
6xy 9y 2
9x 2
5 5 10
2
x 3y
10
2 3
x 3y
10 3x y
10 2 3x y
10 2 10 0
24xy 9y 2
9x 2
10 10 5
2x 2 3xy 2y 2 10 0
6xy y 2
10 0
5 5
5 2 x 2 5 2 y 2 10
x 2
4
y 2
4
1, Hyperbola
9.
5x 2 6xy 5y 2 12 0
A 5, B 6, C 5
cot 2 A C
B
x x cos
4 y sin
4 2
2 x y
0
⇒
y x sin
4 y cos
4 2
2 x y
2
2
⇒
5x 2 6xy 5y 2 12 0
5 2
2
2 x y
6 2
2 x y 2
2 x y
5 2
2
2 x y
12
5
2 x 2 5xy 5 2 y 2 3x 2 3y 2 5 2 x 2 5xy 5 2 y 2 12
4
2 x 2 8y 2 12
x 2
6
y′
−4 −3
y 2
32
4
3
2
−3
−4
y
1, Ellipse
2
x′
3 4
x
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10.
13x 2 63xy 7y 2 16 0
A 13, B 63, C 7
cot 2 A C
B
x x cos
6 y sin
6
x 3
2 y 1 2 x 1 3
2 y 2
3x y
2
—CONTINUED—
1 3 ⇒ 2
3 ⇒
y x sin
6 y cos
6
x 3y
2
6
y
y′
3
−3 −2 2 3
−2
−3
x′
x

812 Chapter 9 Topics in Analytic Geometry
10. —CONTINUED—
39x
4
2
133xy
2
13y 2
18x2
4 4
18y 2
4
7x 2
4
13x 2 63xy 7y 2 16 0
13
3x y
2 2 63
3x y
2 x 3y
2 7 x 3y
2 2 16 0
183xy 63xy
4 4
73xy
2
21y 2
16 0
4
16x 2 4y 2 16
x 2
1
y 2
4
1 ,Ellipse
11.
3x 2 23xy y 2 2x 23y 0
A 3, B 23, C 1
y′
2
y
x′
cot 2 A C
B
1 3 ⇒
60
−6
−4
−2
2
x
x x cos 60 y sin 60
−4
x 1 2
3
y 2 x 3y
2
y x sin y cos 3x y
2
3x 2 23xy y 2 2x 23y 0
3
x 3y
2 2 23
x 3y
2 3x y
2 3x y
2 2 2
x 3y
2
23
3x y
2 0
3(x) 2
4
63xy
4
9(y) 2
4
6(x) 2
4
43xy
4
6(y) 2
4
3(x) 2
4
23xy (y) 2
4 4
x 3y 3x 3y 0
4(y) 2 4x 0
x y
2 ,
Parabola
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Section 9.4 Rotation and Systems of Quadratic Equations 813
12.
16x 2 24xy 9y 2 60x 80y 100 0
y
A 16, B 24, C 9
x′
cot 2
C
B
7 24 ⇒ 53.13
cos 2 7 25
sin
1 cos 2
2
1 725
4 2 5
cos
1 cos 2
2
1 725
3 2 5
x x cos y sin x 3 5 y 4 5 3x 4y
5
y x sin y cos x 4 5 y 3 5 4x 3y
5
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
144x 2
25
384xy 256y 2
288x 2
25 25 25
168xy 288y 2
144x 2
216xy
25 25 25 25
81y 2
25
1
y′
1 2 3 4 5 6
16x 2 24xy 9y 2 60x 80y 100 0
36x 48y 64x 48y 100 0
x
25y 2 100x 100 0
y 2 4x 1
Parabola
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13.
9x 2 24xy 16y 2 90x 130y 0
A 9, B 24, C 16
cot 2 A C
B
cos 2 7
25
sin
1 cos 2
2
1 725
4 2 5
cos
1 cos 2
2
1 725
3 2 5
x x cos y sin
x 3 5 y 4 5
7 24 ⇒
53.13
y x sin y cos
x 4 5 y 3 5
y′
6
4
2
x′
−4 2 4
−2
y
x
3x 4y
5
4x 3y
5
—CONTINUED—

814 Chapter 9 Topics in Analytic Geometry
13. —CONTINUED—
81x 2
25
216xy 144y2
288x2
25 25 25
9x 2 24xy 16y 2 90x 130y 0
9
3x 4y
5 2 24
3x 4y
5 4x 3y
5 16 4x 3y
5 2 90
3x 4y
5
130
4x 3y
5 0
168xy 288y2
256x2
384xy
25 25 25 25
144y2
54x 72y 104x 78y 0
25
25x 2 50x 150y 0
x 2 2x 1 6y 1
x 1 2 6 y 1 6 ,
Parabola
14.
9x 2 24xy 16y 2 80x 60y 0
y
A 9, B 24, C 16
cot 2
C
B
7 24 ⇒ 53.13
y′
3
2
1
x′
cos 2 7 25
sin
1 cos 2
2
1 725
4 2 5
cos
1 cos 2
2
1 725
3 2 5
x x cos y sin y x sin y cos
x x cos y sin x 3 5 y 4 5 3x 4y
5
y x sin y cos x 4 5 y 3 5 4x 3y
5
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
81x 2
25
216xy 144y 2
288x 2
25 25 25
168xy 288y 2
256x 2
384xy
25 25 25 25
144y 2
25
−3 −2 −1 1
−1
9x 2 24xy 16y 2 80x 60y 0
x
48x 64y 48x 36y 0
25x 2 100y 0
x 2 4y,
Parabola
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Section 9.4 Rotation and Systems of Quadratic Equations 815
15.
x 2 3xy y 2 20
cot 2 A C
B
1 1
3
Solve for y in terms of x:
y 2 3xy 20 x 2
0 ⇒
4 45
Graph
y 1 3x
2
80 5x2
2
and
y 2 3x
2
80 5x2
.
2
y 2 3xy 9x2
4 20 x2 9x2
4
y 3 2 x 2 20 5x2
4
80 5x2
4
8
−12 12
−8
y 3 80 5x2
x ±
2 2
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16.
x 2 4xy 2y 2 8 17.
A 1, B 4, C 2
cot 2 A C
B
1
tan 2
1 4
tan 2 4
To graph conic with a graphing calculator, we need
to solve for y in terms of x.
x 2 4xy 2y 2 8
y 2 2xy x 2 4 x2
2 x2
Graph
2 75.96
37.98
y x 2 4 x2
2
y x ± 4 x2
2
y 1 x 4 x2
2
y 2 x 4 x2
2 .
6
−9 9
1 2
4 1 4
y x ± 4 x2
2
and
17x 2 32xy 7y 2 75
cot 2 A C
B
Solve for y in terms of x by completing the square.
y 2 32 7
Graph
7y 2 32xy 17x 2 75
y 2 32
7 xy 17 7 x2 75
7
xy
256
49 x2 119
49 x2 525
49 256
49 x2
y 16
7 x 2 375x2 525
49
y 1 16x 515x2 21
7
y 2 16x 515x2 21
.
7
6
−9 9
−6
17 7
32
24
32 3 4 ⇒
y 16 7 x ± 375x2 525
49
y 16x ± 515x2 21
7
and
26.57
−6

816 Chapter 9 Topics in Analytic Geometry
18.
40x 2 36xy 25y 2 52 19.
A 40, B 36, C 25
cot 2 A C
B
1
tan 2
5 12
tan 2 12 5
2 67.38
33.69
40 25
36
5 12
Solve for y in terms of x by completing the square:
25y 2 36xy 52 40x 2
y 2 36 52
xy
25 25 40
25 x2
y 2 36 324
xy
25 625 x2 52
25 40
25 x2 324
625 x2
y 18
25 x 2 1300 676x2
625
−3 3
y 18
25 x ± 1300 676x2
625
2
−2
32x 2 48xy 8y 2 50
cot 2 A C
B
Solve for y in terms of x:
Graph
6
−9 9
−6
32 8
48
8y 2 48xy 32x 2 50
y 2 6xy 4x 2 25 4
y 2 3x 20x2 25
.
2
1 2 ⇒
y 2 6xy 9x 2 4x 2 25 4 9x2
y 3x 2 5x 2 25 4 20x2 25
4
y 3x ± 20x2 25
2
y 1 3x 20x2 25
2
and
31.72
Graph y 1
y 2
y
18x 1300 676x2
25
18x 1300 676x2
.
25
18x ± 1300 676x2
25
and
20.
4x 2 12xy 9y 2 413 12x 613 8y 91
A 4, B 12, C 9
cot 2 A C
B
1
tan 2
5 12
tan 2 12 5
2 67.38
33.69
—CONTINUED—
4 9
12 5 12
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Section 9.4 Rotation and Systems of Quadratic Equations 817
20. —CONTINUED—
Solve for y in terms of x with the Quadratic Formula:
Graph y 1
y 2
4x 2 12xy 9y 2 413 12x 613 8y 91
9y 2 12x 613 8y 4x 2 413x 12x 91 0
a 9, b 12x 613 8, c 4x 2 413x 12x 91
y b ± b2 4ac
2a
y 12x 613 8 ± 12x 613 82 494x 2 413x 12x 91
18
12x 613 8 ± 624x 3808 9613
18
12x 613 8 624x 3808 9613
18
12x 613 8 624x 3808 9613
.
18
and
18
−9 27
−6
21.
xy 4 0 22.
x 2 2xy y 2 0
B 2 4AC 1 ⇒
The graph is a hyperbola.
x y 2 0
cot 2 A C
B
Matches graph (e).
0 ⇒
45
x y 0
y x
The graph is a line. Matches graph (b).
23.
2x 2 3xy 2y 2 3 0 24.
x 2 xy 3y 2 5 0
B 2 4AC 3 2 422
A 1, B 1, C 3
25 ⇒ The graph is a hyperbola.
B 2 4AC 1 2 413 11
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25.
cot 2 A C
B
Matches graph (f).
4 3 ⇒
18.43
3x 2 2xy y 2 10 0 26.
B 2 4AC 2 2 431
cot 2 A C
B
Matches graph (d).
8 ⇒ The graph is an ellipse or circle.
1 ⇒
22.5
The graph is an ellipse or circle.
cot 2 A C
B
Matches graph (a).
1 3
1
The graph is a parabola.
cot 2 A C
B
Matches graph (c).
1 4
4
2 ⇒
x 2 4xy 4y 2 10x 30 0
A 1, B 4, C 4
B 2 4AC 4 2 414 0
3
4 ⇒
13.28
26.57

818 Chapter 9 Topics in Analytic Geometry
27.
16x 2 24xy 9y 2 30x 40y 0
(a)
B 2 4AC 24 2 4169 0 ⇒
Parabola
(c)
6
(b)
9y 2 24x 40y 16x 2 30x 0
y 24x 40 ± 24x 402 4916x 2 30x
29
−4 8
−2
24x 40 ± 3000x 1600
18
28. (a)
(b)
(c)
B 2 4AC 4 2 412 29.
2y 2 4xy x 2 6 0
y 4x ± 16x2 42x 2 6
4
4x ± 24x2 48
4
8
16 8 24
> 0 ⇒ Hyperbola
15x 2 8xy 7y 2 45 0
(a)
(b)
(c)
B 2 4AC 8 2 4157
7y 2 8xy 15x 2 45 0
y 8x ± 8x2 4715x 2 45
14
8x ±1260 356x2
14
4
356 ⇒ Ellipse or circle
−12 12
−6 6
−8
−4
30. (a)
(b)
(c)
32. (a)
(c)
B 2 4AC 4 2 425 31.
5y 2 4x 4y 2x 2 3x 20 0
y 4 4x ±4x 42 452x 2 3x 20
10
4 4x ±24x2 92x 416
10
−10 8
24 < 0 ⇒ Ellipse or circle
6
−6
B 2 4AC 60 2 43625
0 ⇒ Parabola
−10 2
1
(b)
x 2 6xy 5y 2 4x 22 0
(a)
(b)
(c)
B 2 4AC 6 2 415
5y 2 6xy x 2 4x 22 0
y 6x ± 6x2 45x 2 4x 22
10
6x ±56x2 80x 440
10
−10 8
25y 2 9 60xy 36x 2 0
56 ⇒ Hyperbola
y 60x 9 ± 9 60x2 10036x 2
50
60x 9 ±1080x 81
50
6
−6
© Houghton Mifflin Company. All rights reserved.
−7

Section 9.4 Rotation and Systems of Quadratic Equations 819
33.
x 2 4xy 4y 2 5x y 3 0
(a)
(b)
(c)
B 2 4AC 4 2 414 0
4y 2 4x 1y x 2 5x 3 0
1 4x ±72x 49
8
3
⇒
Parabola
y 1 4x ± 4x 12 44x 2 5x 3
8
34. (a)
(b)
4y 2 x 1y x 2 x 4 0
y x 1 ± x 12 44x 2 x 4
8
(c)
B 2 4AC 1 414
x 1 ±15x2 14x 65
8
2
15 < 0 ⇒ Ellipse or circle
−4 8
−3 3
−5
−2
35.
y 2 16x 2 0
y
36.
y ±4x
Two intersecting lines −3 −2
y 2 16x 2
3
x
x 2 y 2 2x 6y 10 0
x 2 2x 1 y 2 6y 9 10 1 9
Point at 1, 3
x 1 2 y 3 2 0
4
3
2
1
−4 −3 −2 −1 1 2 3 4
y
x
−2
−3
−4
(1, −3)
37.
x 2 2xy y 2 1 0 38.
x 2 10xy y 2 0
x y 2 1 0
y 2 10xy 25x 2 25x 2 x 2
x y 2 1
y 5x 2 24x 2
x y ±1
y 5x ±24x 2
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y x ± 1
Two parallel lines
y
3
2
1
−3 −2 −1 1 2 3
−2
−3
x
Two lines
y 5x ± 26x
y 5 ± 26x
y
4
3
2
1
−2 −1 1 2 3 4
−2
x

820 Chapter 9 Topics in Analytic Geometry
39.
x 2 y 2 4 40.
3x y 2 0
Adding:
x 2 3x 4 0
x 4x 1 0 ⇒ x 1, 4
For x 1, y ±3.
x 4 is impossible.
Solutions: 1, 3, 1, 3
4 27 y 2 9y 2 36y 0
y 36 ± 362 45108
10
36 ± 864
10
No solution
4x 2 9y 2 36y 0
x 2 y 2 27 0 ⇒ x 2 27 y 2
5y 2 36y 108 0
41.
4x 2 y 2 16x 24y 16 0
4x 2 y 2 40x 24y 208 0
24 x 192 0
24 x 192
x 8
For x 8:
464 y 2 168 24y 16 0
y 2 24y 144 0
y 2 24y 144 0
y 12 2 0
⇒ y 12
Solution: 8, 12
42.
x 2 4y 2 20x 64y 172 0 ⇒ x 10 2 4y 8 2 16
16x 2 4y 2 320x 64y 1600 0 ⇒ 16x 10 2 4y 8 2 256
17x 2 340x 1428 0
17x 238x 6 0
x 6 or x 14
When x 6:
When x 14:
6 2 4y 2 206 64y 172 0
14 2 4y 2 2014 64y 172 0
4y 2 64y 256 0
4y 2 64y 256 0
43.
y 2 16y 64 0
y 8 2 0
y 8
Points of intersection: 6, 8, 14, 8
x 2 y 2 12x 16y 64 0
x 2 y 2 12x 16y 64 0
2x 2 24x 0
x 2 12x 0
xx 12 0 ⇒ x 0, 12
y 2 16y 64 0
y 8 2 0
y 8
For x 0:
y 2 16y 64 0
y 2 16y 64 0
y 8 2 0 ⇒ y 8
For x 12:
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144 y 2 1212 16y 64 0
y 2 16y 64 0
⇒
y 8
Solutions: 0, 8, 12, 8

Section 9.4 Rotation and Systems of Quadratic Equations 821
44.
x 2 4y 2 2x 8y 1 0 ⇒ x 1 2 4y 1 2 4
x 2 2x 4y 1 0 ⇒ y 1 4x 1 2
4y 2 12y 0
4yy 3 0
y 0 or y 3
When y 0:
x 2 40 2 2x 80 1 0
x 2 2x 1 0
x 1 2 0
x 1
Point of intersection: 1, 0
When y 3:
x 2 2x 43 1 0
x 2 2x 13 0
No real solution
45.
16x 2 y 2 24y 80 0
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46.
When y 5:
16x 2 255 2 400 0
No real solution
x 0
The point of intersection is 0, 4. In standard form the equations are:
x 2
4
16x 2 25y 2 400 0
y 122
64
24y 2 24y 480 0
24y 5y 4 0
1
x 2
25 y2
16 1
When x 5:
y 5 or y 4
16x 2 225
16x 2 48x 160 0
16x 2 3x 10 0
x 5x 2 0
y 2 485 16y 32 0
y 2 16y 272 0
x 5 or x 2
y 8 ± 421
When y 4:
16x 2 254 2 400 0
16x 2 y 2 16y 128 0 ⇒ 16x 2 y 8 2 64
y 2 48x 16y 32 0 ⇒ y 8 2 48x 96
When x 2:
Points of intersection: 5, 8 421, 5, 8 421, 2, 8
16x 2 0
y 2 482 16y 32 0
y 2 16y 64 0
y 8 2 0
y 8

822 Chapter 9 Topics in Analytic Geometry
47.
2x 2 y 2 6 0 48.
2x y 0 ⇒ y 2x
2x 2 2x 2 6 0
2x 2 6 0
x 2 3 ⇒ x ±3
Two solutions: 3, 23, 3, 23
6x 2 3y 2 12 0
x y 2 0 ⇒ y 2 x
6x 2 32 x 2 12 0
6x 2 34 4x x 2 12 0
9x 2 12x 0
3x3x 4 0
x 0 ⇒ y 2
Solutions: 0, 2, 4 3 , 2 3
x 4 3 ⇒ y 2 4 3 2 3
49.
10x 2 25y 2 100x 160 0 50.
4x 2 y 2 8x 6y 9 0
y 2 2x 16 0 ⇒ y 2 2x 16
2x 2 3y 2 4x 18y 43 0
10x 2 252x 16 100x 160 0
From Equation 1:
10x 2 150x 560 0
y 2 6y 4x 2 8x 9
x 2 15x 56 0
3y 2 18y 12x 2 24x 27
x 8x 7 0
In Equation 2:
x 8 ⇒ y 2 0 ⇒ 8, 0
2x 2 4x 43 3y 2 18y 0
x 7 ⇒ y 2 2
impossible
2x 2 4x 43 12x 2 24x 27 0
One solution: 8, 0
10x 2 28x 16 0
5x 2 14x 8 0
x 25x 4 0
x 2 ⇒ y 2 6y 9
⇒ y 3 2 0 ⇒ y 3
x 4 5 ⇒ y2 6y 321
25
No solution
Solution: 2, 3
51.
xy x 2y 3 0 ⇒ y x 3
x 2
x 2 4y 2 9 0
x 2 4
x 3
x 2 2 9
x 2 x 2 2 4x 3 2 9x 2 2
x 2 x 2 4x 4 4x 2 6x 9 9x 2 4x 4
x 4 4x 3 4x 2 4x 2 24x 36 9x 2 36x 36
x 4 4x 3 x 2 60x 0
x(x 3)x 2 7x 20 0
Note:
x 2 7x 20 0 has no real solution.
When x 0:
When x 3:
y 0 3
0 2 3 2
y 3 3
3 2
0
The points of intersection are 0, 3 2, 3, 0.
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x 0 or x 3

Section 9.4 Rotation and Systems of Quadratic Equations 823
52.
5x 2 2xy 5y 2 12 0
5x 2 2x1 x 51 x 2 12 0
5x 2 2x 2x 2 51 2x x 2 12 0
5x 2 2x 2x 2 5 10x 5x 2 12 0
When x 3 30 :
6
When x 3 30 :
6
Points of intersection:
x y 1 0 ⇒ y 1 x
12x 2 12x 7 0
y 1 3 30
6
y 1 3 30
6
x 3 ±30
6
3 30
6
3 30
6
1 1 6 3 30, 1 6 3 30 6 3 30, 1 6 3 30 ,
53. True. B 2 4AC 1 4k 54. False. See Example 2. However, A C A C.
If k < 1 4 , then B 2 4AC > 0.
55.
gx 2
2 x
Asymptotes:
x 2, y 0
Intercepts: 0, 1
y
4
3
2
(0, 1)
−2 −1 1 3
−2
−3
−4
x
56.
f x
2x
2 x 2 4
2 x
Intercept: 0, 0
Asymptotes:
x 2, y 2
6
4
y
2
(0, 0)
4 6 8
x
−4
−6
© Houghton Mifflin Company. All rights reserved.
57.
ht t 2
2 t t 2 4
2 t
Slant asymptote: y t 2
Vertical asymptote: t 2
Intercept: 0, 0
10
5
−10 −5 5 10 15
−5
−10
−15
y
(0, 0)
t
58.
gs 2
4 s 2
Intercept:
0, 1 2
Asymptotes:
s ±2, y 0
4
3
2
1
−4 −1 1 4
−2
−3
−4
y
( 0, 1 2)
s

824 Chapter 9 Topics in Analytic Geometry
59. (a)
(b)
61. (a)
AB
1
2
BA
0
5
(c) A 2
1
2
(b)
(c)
3
BA 4
A 2
3
5 0 5
6
1 1 2
3
5 1 2
does not exist.
6
1 15
25
3
5 12 3
3
5 5
12
3
4
5
5 4 2
5
AB 4 2 5
30
20
12
16
20
18
19
9
7
12 8 25 45
6
8
10
15
20
25
60. (a)
(b)
AB
1
0
BA
3
3
(c) A 2
1
0
5
2 3
3
2
8 1 0
5
2 1 0
2
8 12 6
5
2 3
3
5
2 1 0
5
4
42
16
11
31
62. (a)
(b)
8
AB 27
13
BA
9 2
9
(c) A 2
2
16
4
10
20
20
14
9
1
2
19
2
2
6
13
63.
0
25
15
10
5
20
f x x 3 64.
10
8
6
4
−8 −6 −4 −2
−2
y
2
4
x
f x x 4 1
10
8
4
2
−2
−2
y
2
4 6 8 10
x
65.
gx 4 x 2 66.
gx 3x 2 67.
ht t 2 3 3
y
y
y
68.
3
1
x
−3 −2 −1 1 2 3
−1
−2
−3
ht 1 2t 4 3 69.
y
4
3
2
1
x
−6 −4 −3 −2 −1
−1
1 2
−2
−3
−4
10
8
6
4
2
−2
−2
2
4 6 8 10
f t t 5 1 70.
3
2
1
−3 −2 −1 1 2 5 6 7
−2
−3
−4
−7
y
x
x
16
14
12
10
8
6
4
2
−10 −8 −6 −4 −2 2 6 8 10
−4
f t 2t 3
−4 −3 −2 −1 1
4
3
2
1
−2
−3
y
2 3 4
x
x
© Houghton Mifflin Company. All rights reserved.

Section 9.5 Parametric Equations 825
71.
73.
Area 1 2 ab sin C 72. Area 1 2ac sin B
1 2812 sin 110
1 22516 sin 70
187.94
45.11
s 1 211 18 10 39
2 74.
Area ss as bs c
39 2 17 2 3 2 19 2
48.60
s 1 223 35 27 85
2
Area ss as bs c
85 2 39 2 15 2 31 2
310.39
Section 9.5
Parametric Equations
■ If f and g are continuous functions of t on an interval I, then the set of ordered pairs ft, gt is a plane
curve C. The equations x ft and y gt 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.
Vocabulary Check
1. plane curve, parametric equations, parameter 2. orientation
3. eliminating, parameter
1.
x t 2.
x t 2 3.
x t
y t 2
y t 2 ⇒ t y 2
y t
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4.
y x 2, line
Matches (c).
x 1 t
⇒ t 1 x
y t 2 ⇒ y 1 x 2
Matches (a).
5.
x y 2 2
Parabola opening to the right
Matches (d).
x ln t ⇔ t e x 6.
y 1 2 t 2
y 1 2 ex 2
Matches (f).
y x 2 , parabola, x ≥ 0
Matches (b).
x
x 2t ⇒ t 2 2 x2
4
y e t
y e x2 4
Exponential curve on x ≤ 0
Matches (e).

826 Chapter 9 Topics in Analytic Geometry
7.
x t, y 2 t
(a)
t
0 1 2 3 4
(c)
3
x
0 1 2 3 2
−4
5
y
2 1 0 1 2
−3
(b) Graph by hand.
Note: x ≥ 0
2
1
−2 −1
1 2
−1
−2
y
x
(d)
y 2 t 2 x 2 ,
Parabola
In part (c), x ≥ 0.
y
1
−2 −1
1 2
−1
−2
x
8.
x 4 cos 2 , y 4 sin
(a)
x
2
4
0
4 2
0 2 4 2 0
(c)
−8
6
10
(b)
y
4
5
4
3
2
1
−5 −4 −3 −2 −1 1 2 3 5
−2
−3
−4
−5
y
22
0 22 4
x
−6
(d)
x
parabola
4 y2
16 1,
The graph is an entire
parabola rather than
just the right portion.
4x y 2 16 cos 2 16 sin 2 16
5
3
2
1
−5 −4 −3 −2 −1 1 2 3 5
−2
−3
y
x
−5
9. The graph opens upward, contains 1, 0, and is 10. The orientation of the graph is clockwise and the
oriented left to right. Matches (b).
center is 2, 3. Matches (c).
11.
x t, y 4t 12.
y 4x
2
−2 −1
1 2
−1
−2
y
x
x t, y 1 2 t
y 1 2 x
−6
−4
6
4
2
−2
−4
or
y
2
x 2y 0
4
6
x
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−6

Section 9.5 Parametric Equations 827
13.
x 3t 3, y 2t 1 14.
x 3 2t, y 2 3t 15.
x 1 4 t, y t 2
t x 3
3
y 2
x 3
3 1
3x 2y 13 0
y
y 2 3
3 x
2
y 4x 2
y 16x 2
y
5
y 2 3 x 3
6
y
5
4
2
1
−6 −4 −3 −2 −1
−1
1 2
x
−4
4
2
−2
−2
−4
2
4
6 8
x
−3 −2 −1
−1
1 2
3
x
−2
−3
16.
x t, y t 3 17.
y x 3
y
6
4
2
x
−6 −4 −2 2 4 6
−6
x t 2, y t 2 18.
t x 2
y x 2 2
4
3
2
1
y
−2 −1 1 2 3 4 5 6
−1
x
2
y
−2
−2
−4
−6
−8
−10
x t
y 1 t
y 1 x 2 , x ≥ 0
2 4
6 8 10
x
19.
x 2t, y t 2
20.
x t 1
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t x 2 ⇒ y t 2
x 2 2
10
8
6
4
−2
−2
y
2 4
1 2 x 4
6 8 10
x
y t 2
Eliminating the parameter t, t y 2 and
x t 1
y 2 1
y 3 .
10
8
6
4
2
−2
−2
y
2
6 8 10
x

828 Chapter 9 Topics in Analytic Geometry
21.
x 2 cos , y 3 sin
22.
x cos , y 4 sin
y
y
4
5
4
2
1
−4 −3 −1 1
−1
3 4
x
−5 −4 −3 −2
2 3 4 5
x
−2
−4
−4
−5
x 2 2 cos 2 , y 3 2 sin 2
x 2
4 y2
9 cos2 sin 2 1
x 2
4 y2
9 1,
ellipse
x 2
y
4 2 cos 2 sin 2 1
x 2 y2
1, ellipse
16
23.
x e t ⇒ 1 x et 24.
y e 3t ⇒ y e t 3
y
1
x 3
y 1 x 3 , x > 0, y > 0
x e 2t
y e t ⇒ y 2 e 2t
y 2 x, y > 0; y x, x > 0
y
5
4
25.
10
9
8
7
6
5
4
3
2
1
y
−2 −1 1 2 3 4 5 6 7 8
y 3 ln t ⇒ y ln t 3
y lnx 13 3
y ln x
5
4
3
2
1
−2 −1 2 3 4 5 6 7 8
−2
−3
−4
−5
y
x
x t 3 ⇒ x 13 t 26.
x
3
2
1
x
1 2 3 4 5
x ln 2t ⇒ e x 2t ⇒ t 1 2 e x
y 2t 2 2 1 2 e x 2 1 2 e2x
y
6
5
4
3
2
1
x
−3 −2 −1 1 2 3
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Section 9.5 Parametric Equations 829
27.
x 4 3 cos , y 2 sin
28.
x 4 3 cos , y 2 2 sin
1
2
−1
8
−2
10
−5
−6
29.
x 4 sec , y 2 tan
30.
x sec
4
8
y tan
−6
6
−12
12
−4
−8
31.
x t 2
8
32.
x 10 0.01e t
35
y lnt 2 1
−12
12
y 0.4t 2
−8
−40
−5
20
33. By eliminating the parameters in (a)–(d), we get y 2x 1. They differ from each other
in restricted domain and in orientation.
(a) Domain: < x <
Orientation: Left to right
(c) Domain: 0 < x <
Orientation: Right to left
(b) Domain: 1 ≤ x ≤ 1
Orientation: Depends on
(d) Domain: 0 < x <
Orientation: Left to right
34. Each curve represents a portion of the line 2y x 8 0.
(a) x 2t, x ≥ 0
(b)
x 2 3 t, < x <
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(c)
y 4 t 4 x 2 , y ≤ 4
Orientation: Left to right
x 2t 1, < x <
y 3 t 3
x 2
2 4 x 2
Orientation: Left to right
(d)
y 4 t 3 4 x 2
Orientation: Left to right
x 2t 2 , x ≤ 0
y 4 t 2 4 x 2
Orientation: Left to right for t ≤ 0
Right to left for t > 0

830 Chapter 9 Topics in Analytic Geometry
35.
t x x 1
x 2 x 1
y y 1
x x 1
x 2 x 1
y 2 y 1
⇒ y y 1
y 2 y 1
x 2 x 1
x x 1
36.
x h r cos
y k r sin
x h
r
cos ,
y k
r
cos 2 sin 2
sin
x h 2
r 2 y k 2
r 2 1
x h 2 y k 2 r 2
37.
x h a cos
38.
x h a sec
y k b sin
y k b tan
x h
a
cos , y k
b
sin
x h
a
sec ,
y k
b
tan
x h 2
a 2
y k2
b 2 1
sec 2 tan 2
x h
2
y k
2
1
a 2 b 2
39.
x x 1 tx 2 x 1 1 t6 1 1 5t 40.
y y 1 ty 2 y 1 4 t3 4 4 7t
x h r cos 2 4 cos
y k r sin 5 4 sin
2, 3 ,
41. a 5, c 4, and b a 2 c 2 3.
42. a 1, c 2, and b c 2 a 2 3.
The center is 0, 0, so h and k 0.
03
The center is 0, 0, so h 0 and k 0.
cos 2 sin 2 1 x2
so x 5 cos and
5 y2
2
y 3 sin . This solution is not unique.
sec 2 tan 2 1 y2
1 x2
x 3 tan .
so y sec and
43.
46.
49.
y 5x 3 44. y 4 7x 45. y 1 x
Answers will vary.
Answers will vary.
Sample answers:
x t, y 5t 3
x t, y 4 7t
x 1 5 t, y t 3
x 2t, y 4 14t
x t, y 1 t
y 1
2x
Sample answers:
x t, y 1 2t
x 2t, y 1 4t
x 2 cot ,
y 2 sin 2
47.
4
y 6x 2 5
Sample answers:
x t, y 6t 2 5
x 2t, y 24t 2 5
50.
x
3t
1 t 3,
y
48.
3t2
1 t 3
x t 3 , y 1 t 3
y x 3 2x
Sample answers:
x t, y t 3 2t
x 1 2 t, y t 3
8 t
4
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−6
6
−6
6
−4
−4

Section 9.5 Parametric Equations 831
51. Matches (b). 52. Matches (c). 53. Matches (d). 54. Matches (a).
55.
x v 0 cos t, y h v 0 sin t 16t 2
(c)
23
(a)
100 mileshour
x 146.67 cos t
100 mihr 5280 ftmi
3600 sechr
146.67 ftsec
x 146.67 cos 23t 135.0t
y 3 146.67 sin 23t 16t 2
3 57.3t 16t 2
60
y 3 146.67 sin t 16t 2
(b)
15
x 146.67 cos 15t 141.7t
0
0
500
y 3 146.67 sin 15t 16t 2
30
3 38.0t 16t 2
(d)
Yes, it is a home run because y > 10 when
x 400.
19.4
is the minimum angle.
0
0
450
It is not a home run because y < 10 when x 400.
56. (a)
x v 0 cos t 105 cos 40t
y h v 0 sin t 16t 2
2.5 105 sin 40t 16t 2
(c) The horizontal distance is approximately
342.25 feet.
(d) You could use the Quadratic Formula to find
the zeros of y 16t 2 105 sin 40t 2.5.
The larger zero, 4.255, gives x 342.25 feet.
(b)
80
0
0
400
The maximum height is approximately
73.68 feet, when t 2.109 seconds.
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57. (a)
(c)
x cos 35v 0 t (b) If the ball is caught at time t 1 , then:
y 7 sin 35v 0 t 16t 2
90 cos 35v 0 t 1
24
0
0
Maximum height 22 feet
(d) From part (b), t 1 2.03 seconds.
90
4 7 sin 35v 0 t 1 16t 12 .
v 0 t 1 90
cos 35 ⇒ 3 sin 35 90
cos 35 16t 1 2
⇒ 16t 12 90 tan 35 3
⇒ t 1 2.03 seconds
⇒ v 0
90
54.09 ftsec
t 1 cos 35

832 Chapter 9 Topics in Analytic Geometry
58. (a)
(b)
x v 0 cos t 85 cos 50t
y h v 0 sin t 16t 2 85 sin 50t 16t 2
80
0
0
240
The maximum height is approximately 66.25 feet when t 2.035 seconds.
(c) The horizontal distance is approximately 222.35 feet.
(d) You could solve the equation y 85 sin 50t 16t 2 0 for t 4.0696.
Then, x 222.35 feet.
59. True
x t first set
y t 2 1 x 2 1
x 3t second set
y 9t 2 1 3t 2 1 x 2 1
60. False. The graph of x t 2 , y t 2 represents the
portion of the line y x in the first quadrant.
61. False. For example, x t 2 and y t does not
represent y as a function of x.
62. False. The equations represent a line.
63. Sample answer:
x cos
y 2 sin
64. The graph is the same, but the orientation is
reversed.
65.
fx
4x2
x 2 1 4x2
Symmetric about the y-axis
Even function
66.
fx x, x ≥ 0
No symmetry
Neither even nor odd
67.
y e x e x ; e x e x
No symmetry
Neither even nor odd
x 2 1 fx 68.
x 2 2 y 4
y x 2 4x
No symmetry
Neither even nor odd
© Houghton Mifflin Company. All rights reserved.

Section 9.6 Polar Coordinates 833
Section 9.6
Polar Coordinates
■ In polar coordinates you do not have unique representation of points. The point r, can be represented
by r, ± 2n or by r, ± 2n 1 where n is any integer. The pole is represented by 0, where
is any angle.
■ To convert from polar coordinates to rectangular coordinates, use the following relationships.
x r cos
y r sin
■ To convert from rectangular coordinates to polar coordinates, use the following relationships.
r ±x 2 y 2
tan yx
If is in the same quadrant as the point x, y, then r is positive. If is in the opposite quadrant as the point
x, y, then r is negative.
■ You should be able to convert rectangular equations to polar form and vice versa.
Vocabulary Check
1. pole 2. directed distance, directed angle 3. polar
1. Polar coordinates:
x 4 cos
4, 2
2 0
y 4 sin 2 4
Rectangular coordinates: 0, 4
2. Polar coordinates:
3
4, 2
x 4 cos
3
2 0, y 4 sin 3
2 4
Rectangular coordinates: 0, 4
© Houghton Mifflin Company. All rights reserved.
3. Polar coordinates:
5
x 1 cos 4 2
2
5
y 1 sin 4 2
2
5
1, 4 4. Polar coordinates:
Rectangular coordinates:
2
2 , 2
2
x 2 cos
y 2 sin
4 2 2
2,
4 2 2
2 2
2 2
Rectangular coordinates: 2, 2
4

834 Chapter 9 Topics in Analytic Geometry
5.
(
3, 5π
6
(
π
2
6.
(
2,
3π
4
(
π
2
1 2 3
0
1 2 3
0
7.
Three additional representations:
5
3, 6 2 3, 7
6
5
3, 6 11
3, 6
5
3,
6 3,
6
π
2
8.
Three additional points:
5
2, 4 , 7
2, 4 ,
2, 4
π
2
9.
( −1, − 3
π
Three additional representations:
1,
1,
1,
3, 5π
6 )
(
3
1 2 3
3 2
3
0
5
1,
3
2
1,
3
1, 4
3
0
1 2 3
Three additional representations:
3, 7
6 , 3,
6 ,
11
3, 6
0
1 2 3 4
−3, −
7π
6 )
Three additional points:
5
3, 6 , 11
3, 6 , 3,
)
)
π
10.
2
0
1 3 5 7 9
6 ,
Three additional points:
52,
π
2
( 5 2, −
11π
6 )
52, 5
6 ,
6
7
52, 6
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Section 9.6 Polar Coordinates 835
11.
π 12.
2
)
)
3
, −
3π
2 2
(
0, −
π
4
(
π
2
1 2 3
0
1 2 3
0
Three additional representations:
3 2 ,
2 , 3 2 , 3
2 , 3 2 , 2
13. Polar coordinates:
x 4 cos
Rectangular coordinates:
π
2
4, 3
3 2
y 4 sin
3 23
2, 23
0, 4
is the origin.
Three additional points:
Any angle will do since r 0.
14. Polar coordinates:
Rectangular coordinates:
π
2
7
2, 6
x 2 cos 7
6 2 3 2 3
y 2 sin 7
6 2 1 2 1
3
0, 4 , 5
0, 4 , 7
0, 4
3, 1
2 4 6
0
1 2 3
0
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15. Polar coordinates:
3
x 1 cos
4
2
2
3
y 1 sin
4
2
2
Rectangular coordinates:
π
2
1 2 3
3
1, 4 16. Polar coordinates:
0
2
2 , 2
2
x 3 cos 2
3 3 1 2 3 2
y 3 sin 2
3 3 3 2 33
2
Rectangular coordinates: 3 2 , 33
2
π
2
1 2 3
3, 2
3 3,
0
3

836 Chapter 9 Topics in Analytic Geometry
5
17. Polar coordinates: 0, 7
(origin!)
6 18. Polar coordinates: 0, (origin!)
4
x 0 cos 5
4 0
x 0 cos 7
6 0
y 0 sin 7
6 0
y 0 sin 5
4 0
Rectangular coordinates:
0, 0
Rectangular coordinates:
0, 0
π
2
π
2
1 2 3
0
1 2 3
0
19. Polar coordinates:
2, 2.36
20. Polar coordinates:
3, 1.57
x 2 cos2.36 1.004
y 2 sin2.36 0.996
x 3 cos1.57 0.0024
y 3 sin1.57 3.000
Rectangular coordinates:
1.004, 0.996
Rectangular coordinates:
0.0024, 3
π
2
π
2
1 2 3
0
1 2 3
0
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:
r 7, tan 0, 0
7, 0
Polar coordinates: 7, , 7, 0
5
4
−9 −8 −7 −6 −5 −4 −3 −2 −1
−2
−3
−4
−5
3
2
1
y
1
x
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Section 9.6 Polar Coordinates 837
30. Rectangular coordinates:
r 5, tan undefined,
2
Polar coordinates:
y
0, 5 31. Rectangular coordinates:
3
5, 2 , 5,
2
Polar coordinates:
y
2,
r 2, tan 1,
4
1, 1
4 ,
5
2,
4
−3
−2
−1
−1
1 2 3
x
3
2
−2
1
−3
− 4
−3
−2
−1
−1
1
2
3
x
−5
−2
−6
−3
32. Rectangular coordinates:
r 32, tan 1,
4
Polar coordinates:
y
3, 3
5
32, 4 , 32,
4
33. Rectangular coordinates:
Polar coordinates:
y
3, 3
r 3 3 6, tan 1,
4
5
6, 4 , 6,
4
2
3
1
2
−4
−3
−2
−1
−1
−2
1 2
x
−3
−2
1
−1
−1
1
2
3
x
−3
−2
−4
−3
34. Rectangular coordinates:
3, 1
35.
x, y 6, 9
r 3 1 2
r 6 2 9 2 117 10.8
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tan 1
3 ,
Polar coordinates:
2
1
y
11
6
−2 −1
−1
1 2 3 4 5 6
−2
−3
−4
−5
−6
11
2, 6 , 5
2, 6
x
tan 9 6 3 2 ⇒
Polar coordinates: 10.8, 0.983, 10.8, 4.124
y
12
9
6
3
x
−3 3 6 9 12
−3
0.983

838 Chapter 9 Topics in Analytic Geometry
36. Rectangular coordinates: 5, 12 37.
r 25 144 13, tan 12 5 ,
1.176
Polar coordinates: 13, 1.176, 13, 4.318
x, y 3, 2 ⇒ r 3 2 2 2 13
r, 13, 0.588
arctan 2 3 0.588
y
12
10
8
6
4
2
−6
−4
−2
−2
2
4 6 8
x
38. x, y 5, 2 ⇒ r, 5.39, 2.76
40. x, y 32, 32 ⇒ r, 6,
4 6.0, 0.785
39.
x, y 3, 2 ⇒ r 3 2 2 7
r, 7, 0.857
arctan
2
3 0.857
41.
x, y
5
2 , 4 3 ⇒ r 5 2 2
4
3 2 17 6
r,
17
6 , 0.490
arctan
43
52 0.490
42. x, y
7
4 , 3 2 ⇒ r, 2.30, 0.71
43.
46.
49.
x 2 y 2 9 44.
r 2 9
r 3
x 2 y 2 16 45.
r 2 16
r 4
y x 47. x 8
48.
r sin r cos
sin cos
tan 1
4
r cos 8
r 8 sec
3x 6y 2 0 50.
3r cos 6r sin 2
r3 cos 6 sin 2
r
2
6 sin 3 cos
y 4
r sin 4
r 4 csc
x a
r cos a
4x 7y 2 0
4r cos 7r sin 2 0
r4 cos 7 sin 2
r
r a sec
2
4 cos 7 sin
© Houghton Mifflin Company. All rights reserved.

Section 9.6 Polar Coordinates 839
51.
xy 4 52.
r cos r sin 4
r 2 cos sin 4
r 2 2 cos sin 8
r 2 sin 2 8
r 2 8 csc 2
2xy 1
2r cos r sin 1
2r 2 sec csc
r 2 1 2 sec csc csc2
53.
x 2 y 2 2 9x 2 y 2 54.
r 2 2 9r 2 cos 2 r 2 sin 2
r 2 9cos 2 sin 2
r 2 9 cos2
r
r 2 sin 2 8r cos 16 0
r 2 1 cos 2 8r cos 16 0
r 2 cos 2 8r cos 16 r 2
4
1 cos
y 2 8x 16 0
r cos 4 2 r 2
or r 4
1 cos
r ±r cos 4
55.
x 2 y 2 6x 0 56.
x 2 y 2 8y 0 57.
x 2 y 2 2ax 0
r 2 6r cos 0
r 2 8r sin 0
r 2 2ar cos 0
r 2 6r cos
rr 8 sin 0
rr 2a cos 0
r 6 cos
r 8 sin
r 2a cos
58.
x 2 y 2 2ay 0 59.
y 2 x 3 60.
x 2 y 3
r 2 2a r sin 0
r sin 2 r cos 3
r 2 cos 2 r 3 sin 3
rr 2a sin 0
r 2a sin
sin 2 r cos 3
r sin2
cos 3
r cos2
cot
sin 3
2 csc
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61.
r 6 sin
r 2 6r sin
x 2 y 2 6y
x 2 y 2 6y 0
62.
x 2 y 2 2x
tan 2 sec
r 2 cos
r 2 2r cos
63.
4
tan tan 4
3 y x
3 y x
3
y 3x

840 Chapter 9 Topics in Analytic Geometry
64.
5
3
65.
5
6
66.
11
6
tan tan 5
3 3
tan tan 5
6 y x
tan tan 11
6 y x
y
x 3
3
3
y x
3
3
y x
y 3x 0
y 3 x
3
y 3 x
3
67. , vertical line
68. horizontal line
2
y 0
x 0
,69.
r 4
r 2 16
x 2 y 2 16
70.
r 10 71.
r 3 csc
72.
r 2 sec
r 2 100
r sin 3
r cos 2
x 2 y 2 100
y 3
x 2
73.
r 2 cos
74.
r 2 sin 2 2 sin cos
r 3 r cos
x 2 y 2 32 x
x 2 y 2 x 23
x 2 y 2 3 x 2
y
r 2 2 r x r 2xy
r 2
r 4 2xy
x 2 y 2 ) 2 2xy
75.
r 2 sin 3
76.
r 3 cos 2
r 23 sin 4 sin 3
r 3cos 2 sin 2
r 4 6r 3 sin 8r 3 sin 3
r 3 3r 2 cos 2 r 2 sin 2
x 2 y 2 2 6x 2 y 2 y 8y 3
x 2 y 2 32 3x 2 y 2 or x 2 y 2 3 9x 2 y 2 2
77.
x 2 y 2 2 6x 2 y 2y 3
r
r r cos 1
x 2 y 2 x 1
1
1 cos
x 2 y 2 1 2x x 2
y 2 2x 1
78.
r
r r sin 2
x 2 y 2 y 2
x 2 y 2 2 y 2
x 2 y 2 4 4y y 2
x 2 4y 4 0
2
1 sin
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Section 9.6 Polar Coordinates 841
79.
r
6
2 3 sin
80.
r
6
2 cos 3 sin
r2 3 sin 6
2±x 2 y 2 6 3y
4x 2 y 2 6 3y 2
4x 2 4y 2 36 36y 9y 2
4x 2 5y 2 36y 36 0
2r 6 3r sin
r
r
1
2 x 3y 6
6
2xr 3yr
6r
2x 3y
6
2x 3y
81.
y
r 7 82.
r 2 49
x 2 y 2 49
The graph is a circle
centered at the origin
with radius 7.
8
6
4
2
−8 −4 −2
−2
2 4 6 8
−4
−6
−8
x
r 8
r 2 64
x 2 y 2 64
Circle of radius 8
centered at origin
−10
10
6
4
2
−6 −4 −2
−4
−6
−10
y
2
4
6 10
x
83.
4
tan tan
4 1 y x
y x
The graph is the line
y x, which makes an
angle of with
the positive x-axis.
4
−3
−2
3
2
1
−1
−1
−2
−3
y
1
2
3
x
84.
7
y
x tan tan 7
6 3
3
3y 3x 0
Line through origin making
angle of 6 with positive
x-axis
6
3
2
1
−3 −2
1 2 3
−1
−2
−3
y
x
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85.
r 3 sec
r cos 3
x 3
x 3 0
Vertical line
−2
3
2
1
−1
−1
−2
−3
y
1
2 4
x
86.
r 2 csc
r sin 2
y 2
y 2 0
Horizontal line through
0, 2
87. True, the distances from the origin are the same. 88. False. For instance when r 0, any value of
gives the same point.
−3
−2
4
3
1
−1
−1
−2
y
1
2
3
x

842 Chapter 9 Topics in Analytic Geometry
89. (a) r where x 1 r 1 cos and y 1 r 1 sin 1 .
1 , 1 x 1 , y 1
1
r where x 2 r 2 cos and y 2 r 2 sin 2 .
2 , 2 x 2 , y 2
2
Then x 12 y 12 r
2
1 cos 2 1 r 1 2 sin 2 1 r 1 2 and x 22 y 22 r 22 . Thus,
d x 1 x 2 2 y 1 y 2 2
x 12 2x 1 x 2 x 22 y 12 2y 1 y 2 y
2
2
x 12 y 12 x 22 y 22 2x 1 x 2 y 1 y 2
r 12 r 22 2r 1 r 2 cos 1 cos 2 r 1 r 2 sin 1 sin 2
r 12 r 22 2r 1 r 2 cos1 2.
(b) If
(c) If
1
1
2, the points are on the same line through the origin. In this case,
d r 12 r 22 2r 1 r 2 cos0 r 1 r 2 2 r 1 r 2 .
2 90, d r 1 2 r 22 , the Pythagorean Theorem.
7
(d) For instance, gives 3, gives d 2.053. (Same!)
6 , 4
3,
d 2.053 and 4, 3
90. Answers will vary.
6 , 4, 3
91.
cos A b2 c 2 a 2
2bc
A 30.7
cos B a2 c 2 b 2
2ac
B 48.2
192 25 2 13 2
21925
132 25 2 19 2
21325
C 180 30.7 48.2 101.1
0.86 92.
0.66615
A 24, a 10, b 6
sin B b sin A
a
C 180 A B 141.9
c a sin C
sin A 15.17
0.2440 ⇒ B 14.1
93.
95.
B 180 56 38 86 94.
a
sin A
b
sin B
c
sin C ⇒ a c sin A
sin C
c
sin C ⇒ b c sin B
sin C
12 sin56
16.16
sin38
12 sin86
19.44
sin38
c 2 a 2 b 2 2ab cos C 96.
8 2 4 2 284 cos35
27.57
c 5.25
b
sin B c
sin C
⇒ sin B
b sin C
c
⇒ B 25.9
A 180 B C 119.1
4 sin35
5.25
B 71, a 21, c 29
b 2 a 2 c 2 2ac cos B 885.458 ⇒ b 29.76
sin C c sin B
b
A 180 B C 41.9
B 64, b 52, c 44
sin C c sin B
b
A 180 B C 66.5
a b sin A
sin B 53.06
0.9214 ⇒ C 67.1
0.7605 ⇒ C 49.5
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Section 9.6 Polar Coordinates 843
97. By Cramer’s Rule,
98.
5 7
1
D 5 21 16
3
D x 11 7
1
11 21 32
3
5 11
D y 15 33 48
3 3 x D x
, y Dy
D 48
D 32
16 2
16 3
Solution: 2, 3
By Cramer’s Rule,
D 3 5
6 20 26
4 2
D x 10 5
20 25 5
5 2
D y 3 10
15 40 55
4 5
x D x
,
D 5 26 ,
Solution: 5 26 , 55
26
y Dy
D 55
26 55
26
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99. By Cramer’s Rule,
9
3 2 1
D 2 1 3 35
1 3
9
0 2 1
D a 0 1 3 0
0 3
9
3 0 1
D b 2 0 3 0
1 0
0
3 2 0
D c 2 1 0 0
1 3
a D a
, b D b
, c D c
D 0 D 0 D 0
Solution: 0, 0, 0
100. By Cramer’s Rule,
1
5 7 9
D 1 2 3 89
8 2
1
15 7 9
D u 7 2 3 295
0 2
1
5 15 9
D v 1 7 3 844
8 0
0
5 7 15
D w 1 2 7 672
8 2
u D u
v D v
D 844
89 844
D 295
89 295
89 ,
89 ,
w D w
D 672
89 672
89
Solution:
295
89 , 844
89 , 672 89

844 Chapter 9 Topics in Analytic Geometry
101. By Cramer’s Rule,
1
D 2
5
1
D x 2
4
1
D y 2
5
D z
1
2
5
1
3
4
1
3
4
1
2
4
1
3
4
x D x
y D y
D 45
D 30
15 2, 15 3,
z D z
D 45
15 3
Solution: 2, 3, 3
2
2
1 15
2
2
1 30
2
2
1 45
4
1
2 45
102. Cramer’s Rule does not apply because
6
2 1 2
D 2 2 0 0.
2 1
Use elimination to solve the system.
2 1 2 4
2 2 0 5
2 1 6 2 →
R 1 R 2
R 1 R 3
2 0
0
R
2 2 R 1
0
2R 2 R 3 0
x 3 a,
1
1
2
0
1
0
2
2
4
4
2
0
4
1
2 →
3
1
0
Let then x and x 1 2a 3 2 2a 1
2 .
Solution: 2a 3 2 , 2a 1, a
103. Points:
1
4 3 1
6 7 1 20 0
2 1
The points are not collinear.
4, 3, 6, 7, 2, 1 104. Points:
2
0
4
2, 4, 0, 1, 4, 5
4
1
5
1
1
1
0 ⇒ collinear
105. Points: 6, 4, 1, 3, 1.5, 2.5
1
6 4 1
1 3 1 0
1.5 2.5
The points are collinear.
106. Points:
2.3
0.5
1.5
2.3, 5, 0.5, 0, 1.5, 3
5
0
3
1
1
1
4.6 ⇒ not collinear
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Section 9.7 Graphs of Polar Equations 845
Section 9.7
Graphs of Polar Equations
■ When graphing polar equations:
1. Test for symmetry
2:
(a) Replace r, by r, or r, .
(b) Polar axis: Replace r, by r, or r, .
(c) Pole: Replace r, by r, or r, .
(d) r f sin is symmetric with respect to the line
(e) r f cos is symmetric with respect to the polar axis.
r
2. Find the values for which is maximum.
3. Find the values for which r 0.
2.
4. Know the different types of polar graphs.
(a) Limaçons (b) Rose curves, n ≥ 2 (c) Circles (d) Lemniscates
r a ± b cos r a cos n
r a cos r 2 a 2 cos 2
r a ± b sin r a sin n
r a sin r 2 a 2 sin 2
r a
■ You should be able to graph polar equations of the form r f with your graphing utility. If your utility
does not have a polar mode, use
x f t cos t
y f t sin t
in parametric mode.
Vocabulary Check
1. 2. polar axis 3. convex limaçon
2
4. circle 5. lemniscate 6. cardioid
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1. r 3 cos 2 is a rose curve.
4. r 3 cos is a circle.
2. Cardioid
7. The graph is symmetric about the line and
passes through r, 3, 32. Matches (a).
5. r 6 sin 2 is a rose curve.
3. Lemniscate
6. Limaçon
2, 8. The graph is symmetric about the polar axis and
passes through r, 3, 0. Matches (c).
9. The graph has four leaves. Matches (c). 10. The graph has three leaves. Matches (d).

846 Chapter 9 Topics in Analytic Geometry
11.
r 14 4 cos
2 :
r 14 4 cos
12.
r 12 cos 3
2 :
r 12 cos3
r 14 4 cos
r 12 cos 3
Not an equivalent equation
Not an equivalent equation
r 14 4 cos
r 14 4cos cos sin sin
r 12 cos3
r 12 cos 3
r 14 4 cos
Not an equivalent equation
Polar axis:
Not an equivalent equation
r 14 4 cos
Polar axis:
r 12 cos3
r 12 cos 3
r 14 4 cos
Equivalent equation
Equivalent equation
Pole:
r 12 cos 3
Pole:
r 14 4 cos
Not an equivalent equation
Not an equivalent equation
r 14 4 cos
r 12 cos3
r 12 cos 3
r 14 4 cos
Not an equivalent equation
Not an equivalent equation
Answer: Symmetric with respect to polar axis
Answer: Symmetric with respect to polar axis
13.
r
4
1 sin
2 :
r
4
1 sin
Pole:
r
4
1 sin
Polar axis:
r
r
Equivalent equation
r
r
Not an equivalent equation
r
r
4
1 sin cos cos sin
4
1 sin
4
1 sin
4
1 sin
4
1 sin
4
1 sin
Not an equivalent equation
Not an equivalent equation
r
r
4
1 sin
4
1 sin
Not an equivalent equation
Answer: Symmetric with respect to
2
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Section 9.7 Graphs of Polar Equations 847
14.
r
2
1 cos
2 :
Polar axis:
r
r
Not an equivalent equation
r
r
r
Not an equivalent equation
r
r
2
1 cos
2
1 cos
2
1 cos
2
1 cos cos sin sin
2
1 cos
2
1 cos
2
1 cos
Equivalent equation
Pole:
2
r
1 cos
Not an equivalent equation
r
r
r
2
1 cos
2
1 cos cos sin sin
2
1 cos
Not an equivalent equation
Answer: Symmetric with respect to the polar axis
15.
r 6 sin
16.
r 4 csc cos 4 cot
2 :
r 6 sin
2 :
r 4 cot
r 6 sin
r 4 cot
Equivalent equation
Equivalent equation
Polar axis:
r 6 sin
Polar axis:
r 4 cot
r 6 sin
r 4 cot
Not an equivalent equation
r 4 cot
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r 6 sin
r 6sin cos cos sin
r 6 sin
Not an equivalent equation
Pole: r 6 sin
Not an equivalent equation
r 6 sin
r 6 sin
Not an equivalent equation
Answer: Symmetric with respect to
2
Equivalent equation
Pole: r 4 cot
r 4 cot
Equivalent equation
Answer: Symmetric with respect to
polar axis and pole
2 ,

848 Chapter 9 Topics in Analytic Geometry
17.
r 2 16 sin 2
18.
r 2 25 cos 4
2 :
r 2 16 sin2
2 :
r 2 25 cos4
r 2 16 sin 2
r 2 25 cos 4
Not an equivalent equation
Equivalent equation
r 2 16 sin2
Polar axis:
r 2 25 cos4
r 2 16 sin2 2
r 2 25 cos 4
r 2 16 sin 2
Equivalent equation
Not an equivalent equation
Pole:
r 2 25 cos 4
Polar axis:
r 2 16 sin2
r 2 25 cos 4
r 2 16 sin 2
Equivalent equation
Not an equivalent equation
r 2 16 sin2
r 2 16 sin 2
Answer: Symmetric with respect to
polar axis and pole
2 ,
Not an equivalent equation
Pole:
r 2 16 sin2
r 2 16 sin 2
Equivalent equation
Answer: Symmetric with respect to pole
19.
r 101 sin 20.
10 1 sin ≤ 102 20
r 6 12 cos ≤ 6 12 cos
6 12 cos ≤ 18
1 sin 2
1 sin 2 or 1 sin 2
cos 1
0
sin 1 sin 3
Maximum:
3
r 20
when
r 0 when 1 sin 0
sin 1
2
or Not possible
2
3
2
Maximum:
r 18 when 0
Zero: r 0 when 2
3 , 4
3
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Section 9.7 Graphs of Polar Equations 849
21.
r 4 cos 3 4 cos 3 ≤ 422. r sin 2
cos 3 1
r sin 2
cos 3 ±1
Maximum: r 1 when
Maximum:
0,
3 , 2
3 ,
r 4
when
r 0 when cos 3 0
5
6 ,
2 , 6
0,
3 , 2
3 ,
4 , 3
4 , 5
4 , 7
4
3
Zero: r 0 when 0,
2 , ,
2 , 2
23.
r 5 24.
Circle
π
2
5
Line
3
π
2
25.
r 3 sin
Symmetric with respect to
2
Circle with radius of 32
π
2
2 4 6 8
0
1 2
0
1 2 3
0
26.
r 2 cos
27.
r 31 cos 28.
r 41 sin
Circle
Radius: 1, center:
1, 0
Cardioid
π
2
Cardioid
π
2
π
2
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29.
1 3
r 3 4 cos
Limaçon
π
2
0
30.
2 4
r 1 2 sin
Limaçon with inner loop
π
2
1 2
0
0
31.
r 4 5 sin
Limaçon
π
2
4 6 8
0
1 2
0
4 6 8
0

850 Chapter 9 Topics in Analytic Geometry
32.
r 3 6 cos
33.
r 5 cos 3
34.
r sin 5
Limaçon
Rose curve
Rose curve
π
2
π
2
π
2
4 6 8 10 12
0
1 5 6
0
1
0
35.
r 7 sin 2
36.
r 3 cos 5
37.
r 8 cos 2
Rose curve, four petals
Rose curve, five petals
12
π
2
π
2
−18
18
−12
4 5 6
0
1 2 3 4
0
0 ≤
< 2
38.
2
39.
r 25 sin
40.
4
−3
3
10
−12
12
−18
18
−2
−12
0 ≤
≤ 2
0 ≤
−14
< 2
0 ≤
≤ 2
41.
44.
r 3 6 cos ,42.
0 ≤ ≤ 2
6
−14
4
−6
4
−6
6
−4
0 ≤
≤ 2
45.
2
−6
6
−6
0 ≤
≤ 2
43.
r 2 4 cos 2,46.
≤ ≤ 2
2
2
−3
3
−2
r
−6
3
,
sin 2 cos
4
−4
r 2 9 sin
r ±3sin
6
0 ≤
Graph both functions using
0 ≤ ≤ 2.
4
≤
2
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−6
6
−4

Section 9.7 Graphs of Polar Equations 851
47.
r 4 sin cos 2 , 0 ≤
≤
48.
2
49.
r 2 csc 6
2
−3
3
14
−3
3
−2
−18
18
−2
0 ≤
≤
0 ≤
−10
< 2
50.
12
51.
r e 2
52.
r e2
−18
18
−2000
200
400
15
−15 30
−12
0 ≤
≤ 2
−1400
−15
Answers will vary.
Answers will vary.
53.
r 3 2 cos ,54.
0 ≤ < 2
4
−6
1
6
55.
r 2 cos
3
2 , 0 ≤
2
< 4
−6
6
−7
−3
3
−4
0 ≤
< 2
−2
56.
−6
4
6
57.
r 2 sin 2,0≤
<
Use r 1 sin 2 and r 2 sin 2.
2
1
−4
0 ≤
< 4
−1
1
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58.
−1.5
1
−1
1.5
0 <
r ±1
<
59.
−1
r 2 sec
x 1 is an asymptote.
4
−6
6
−4

852 Chapter 9 Topics in Analytic Geometry
60.
r sin 2 sin 1
rr sin 2r sin r
±x 2 y 2 y 2y ±x 2 y 2
±x 2 y 2 y 1 2y
±x 2 y 2
r 2 csc 2 1
sin
2y
y 1
x 2 y 2 4y 2
y 1 2
x 2 y 2 3 2y y 2
y 1 2
x ± y 2 3 2y y 2
y 1 2
The graph has an asymptote at y 1.
±
5
−6
6
−3
y
2y y
y 13 2
61.
r 2
3
62.
4
y 2 is an asymptote.
−3
3
−6
6
−1
−4
63. True. It has five petals. 64. False. For example, let r cos 3.
65. r cos5 n cos , 0 ≤ < ; Answers will vary.
4
4
4
2
−6
6
−6
6
−6
6
−3
3
n 5
−3
n 1
−6
−4
2
−2
4
3
6
n 4
−3
n 0
−6
−4
2
−2
4
3
6
n 3
−3
n 1
−6
−4
2
−2
4
3
6
n 2
−3
n 2
−2
2
−2
3
© Houghton Mifflin Company. All rights reserved.
−4
−4
−4
n 3
n 4
n 5

Section 9.7 Graphs of Polar Equations 853
66. The graph of r f is rotated about the pole through an angle . Let r, be any point
on the graph of r f . Then r, is rotated through the angle , and since
r f f , it follows that r, is on the graph of r f .
67. Use the result of Exercise 66.
(a) Rotation:
2
Original graph: r f sin
Rotated graph:
(b) Rotation:
Original graph:
Rotated graph:
(c) Rotation:
3
Original graph:
r f sin
r f sin
2
r f sin
Rotated graph: r f sin 3
2 f cos
2 f cos
r f sin f sin
68. (a)
(b)
(c)
(d)
r 2 sin
r 2 sin
r 2 cos
r 2 sin
r 2 cos
4
2 2
2 sin cos
2
r 2 sin
r 2 sin 3
2
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69. (a)
(b)
(c)
(d)
r 2 sin 2
2 sin 2
sin 2 3 cos 2
r 2 sin 2
2 sin2
2 sin 2
4 sin cos
r 2 sin 2 2
3
2 sin 2 4
3
3 cos 2 sin 2
r 2 sin2
2 sin2 2
2 sin 2
4 sin cos
6
3
2
70. (a)
(b)
r 1 sin
y
1
–2 1 2
–1
–3
r 1 sin
y
1
–2 –1 1 2
–2
–3
4
x
x

854 Chapter 9 Topics in Analytic Geometry
71.
r 2 k cos
k 0
Circle
k 1
Convex limaçon
k 2
Cardioid
k 3
Limaçon with inner loop
4
4
4
4
−6
6
−6
6
−4
8
−4
8
−4
−4
−4
−4
72.
r 3 sin k
(a)
4
(b)
4
(c) Yes. Answers will vary.
−6
6
−6
6
−4
−4
k 1.5: 0 ≤
< 4
k 2.5: 0 ≤
< 4
Section 9.8
Polar Equations of Conics
■
■
The graph of a polar equation of the form
r
ep
1 ± e cos
Vocabulary Check
or
r
ep
1 ± e sin
is a conic, where e > 0 is the eccentricity and is the distance between the focus (pole) and the directrix.
(a) If e < 1, the graph is an ellipse.
(b) If e 1, the graph is a parabola.
(c) If e > 1, the graph is a hyperbola.
Guidelines for finding polar equations of conics:
(a) Horizontal directrix above the pole:
(b) Horizontal directrix below the pole:
r
r
(c) Vertical directrix to the right of the pole:
r
(d) Vertical directrix to the left of the pole: r
ep
1 e sin
ep
1 e sin
ep
1 e cos
ep
1 e cos
1. conic 2. eccentricity, e
3. (a) i (b) iii (c) ii
p
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Section 9.8 Polar Equations of Conics 855
1.
2e
r
1 e cos
(a) Parabola
(b) Ellipse
(c) Hyperbola
−4
a
b
4
−4
c
8
2. (a) Parabola
(b) Ellipse
(c) Hyperbola
−8
c
4
−4
a
b
4
3.
2e
r
1 e sin
(a) Parabola
(b) Ellipse
(c) Hyperbola
−9
4
b a
c
−8
9
4. (a) Parabola
(b) Ellipse
(c) Hyperbola
−9
8
c
b
a
−4
9
5.
r
4
1 cos
6.
r
3
32
2 cos 1 12 cos
e 1 ⇒ parabola
Vertical directrix to left of pole
Matches (b).
e 1 2 ⇒
ellipse
Vertical directrix to left of pole
Matches (c).
7.
r
3
32
2 cos 1 12 cos
8.
r
4
1 3 sin
9.
r
3
1 2 sin
e 1 2 ⇒
ellipse
Vertical directrix to right of pole
e 3 ⇒ hyperbola
Horizontal directrix below
the pole.
e 2 ⇒ hyperbola
Horizontal directrix above
the pole.
Matches (f).
Matches (e).
Matches (d).
10.
r
4
1 sin
11.
r
2
1 cos
12.
r
2
1 sin
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13.
e 1 ⇒ parabola
Vertex: 2, 2
Matches (a).
r
4
1
4 cos 1 14 cos
e 1 ellipse
4 , p 4, 0
Vertices:
r,
4
3 , 0 , 4 5 ,
e 1 ⇒ parabola
Vertex: r, 1,
π
2
1 2
e 1 ⇒ parabola
Vertex: 1, 2

856 Chapter 9 Topics in Analytic Geometry
14.
r
7
1
7 sin 1 17 sin
15.
r
8
2
4 3 sin 1 34 sin
e 1 7 ⇒ ellipse
e 3 4 ⇒ ellipse
π
2
Vertices: r,
7
8 ,
2 , 7 6 , 3
2
Vertices:
r,
8
7 ,
2 ,
3
8,
2
1 3 4 5 6
0
16.
r
6
2
3 2 cos 1 23 cos
17.
r
6 126
2 sin 1 12 sin
e 2 ellipse
3 ⇒
e 1 ellipse
2 ⇒
π
2
Vertices: 6, 0,
6
5 ,
Vertices:
2,
2 ,
3
6,
2
1 2 4 5
0
18.
5
5
r
1 2 cos 1 2 cos
e 2 ⇒ hyperbola
Vertices: 5, 0, 5 3 ,
19.
3
34
r
4 8 cos 1 2 cos
e 2 ⇒ hyperbola
Hyperbola
Vertices:
r, 3 1 4 , 0 , 4 , 0
1 2 3 4 5
π
2
20.
23.
10
r
103
3 9 sin 1 3 sin
e 3 ⇒ hyperbola
Vertices:
r,
5
6 ,
r
14
14 17 sin
2 , 5 3 , 3
1
1 1714 sin
2
−9
21.
9
r
5
1 sin
Parabola
−6
9
4
−4
24.
6
12
r
2 cos
Ellipse
22.
1
r
2 4 sin
Hyperbola
−3
−15
2
−2
10
3
15
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Hyperbola
−3
−10

Section 9.8 Polar Equations of Conics 857
25.
2
26.
2 27.
3
−4
2
−5
1
−2
4
−2
−2
−
Ellipse
Hyperbola
28.
2
29.
2 30.
3
−3
3
−3
3
−9
9
−2
−2
−9
31.
4
32.
−8
1
4
−9
9
−8
−7
33.
e 1, x 1, p 1 34.
Vertical directrix to the left of the pole
r
11 1
1 1 cos 1 cos
e 1, y 4, p 4
Horizontal directrix below the pole
r
14 4
1 1 sin 1 sin
35.
e 1 2 , y 1, p 1
36. e 3 4 , y 4, p 4
Horizontal directrix below pole
Horizontal directrix above the pole
r
121 1
1 12 sin 2 sin
(344
12
r
1 34 sin 4 3 sin
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37. e 2, x 1, p 1 38. e 3 2 , x 1, p 1
Vertical directrix to the right of the pole
Vertical directrix to the left of the pole
21
2
r
321
3
1 2 cos 1 2 cos
r
1 32 cos 2 3 cos
39. Vertex: 1, ⇒ e 1, p 2 40. Parabola, e 1, vertex: 8, 0
2
Vertical directrix to right of pole
Horizontal directrix below the pole
ep 16
12 2
r
r
1 e cos 1 cos
1 1 sin 1 sin

858 Chapter 9 Topics in Analytic Geometry
41. Vertex: 5, ⇒ e 1, p 10
Vertical directrix to left of pole
r
43. Center: 4, , c 4, a 6, e 2 3
Vertical directrix to the right of the pole
r
2
r
45. Center:
23p
2p
1 23 cos 3 2 cos
2p
3 2 cos 0 2p 5 ⇒ p 5
10
3 2 cos
Vertical directrix to left of pole
r
20 2p
3 2
r
110 10
1 1 cos 1 cos
8, 0, c 8, a 12, e c a 2 3
23p
2p
1 23 cos 3 2 cos
20
3 2 cos
2p ⇒ p 10
42. Vertex:
Horizontal directrix above pole
r
44. Center:
Horizontal directrix above the pole
r
2
p 8
r
46. Center:
3
1, 2 , c 1, a 3, e 1 3
13 p p
1 13 sin 3 sin
p
3 sin2
8
3 sin
Horizontal directrix below the pole
r
1
p 9 5
10,
2
3
5, 2 , c 5, a 4, e 5 4
54p
5p
1 54 sin 4 5 sin
5p
4 5 sin32
⇒ e 1, p 20
120 20
1 1 sin 1 sin
47. Center: 5 2 , 2 , c 5 2 , a 3 2 , e 5 3
Horizontal directrix above the pole
r
Substitute the point
rather than
in order to get a directrix between the vertices.
1
p 8 5
r
53p
5p
1 53 sin 3 5 sin
5p
3 5 sin32
3
1, 2
585 8
3 5 sin 3 5 sin
3
1, 2
r
595 9
4 5 sin 4 5 sin
48. Center: 5 2 , 2 , c 5 2 , a 3 2 , e c a 5 3
Horizontal directrix above the pole
r
1
r
53p
5p
1 53 sin 3 5 sin
5p
3 5 sin2 ⇒ p 8 5
8
3 5 sin
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Section 9.8 Polar Equations of Conics 859
49. When
Therefore,
Thus, r
0, r c a ea a a1 e. 50. Minimum distance occurs when .
a1 e
a1 e1 e ep
a1 e 2 ep.
ep
1 e cos 0
ep
1 e2 a
.
1 e cos 1 e cos
r
1 e2 a 1 e1 ea
a1 e
1 e cos 1 e
Maximum distance occurs when
0.
r 1 e2 a 1 e1 ea
a1 e
1 e cos 0 1 e
51.
r 1 0.01672 92.956 10 6
1 0.0167 cos
9.2930 107
1 0.0167 cos
Perihelion distance:
r 92.956 10 6 1 0.0167 9.1404 10 7
Aphelion distance:
r 92.956 10 6 1 0.0167 9.4508 10 7
52.
a 35.983 10 6 , e 0.2056
r 1 0.20562 35.983 10 6
1 0.2056 cos
3.4462 107
1 0.2056 cos
Perihelion distance: a1 e 2.8585 10 7 miles
Aphelion distance: a1 e 4.3381 10 7 miles
53.
r 1 0.04842 77.841 10 7
1 0.0484 cos
7.7659 108
1 0.0484 cos
Perihelion:
r 77.841 10 7 1 0.0484 7.4073 10 8 km
Aphelion:
r 77.841 10 7 1 0.0484 8.1609 10 8 km
54.
a 142.673 10 7 , e 0.0542
r 1 0.05422 142.673 10 7
1 0.0542 cos
1.4225 109
1 0.0542 cos
Perihelion distance: a1 e 1.3494 10 9 km
Aphelion distance: a1 e 1.5041 10 9 km
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55. a 4.498 10 9 , e 0.0086, Neptune
a 5.906 10 9 , e 0.2488, Pluto
(a) Neptune:
Pluto: r 1 0.24882 5.906 10 9 5.5404 109
1 0.2488 cos 1 0.2488 cos
(b) Neptune: Perihelion: 4.498 109 1 0.0086 4.4593 10 9 km
Aphelion:
Pluto: Perihelion:
r 1 0.0086 2 4.498 10 9
1 0.0086 cos
4.498 10 9 1 0.0086 4.5367 10 9 km
5.906 10 9 1 0.2488 4.4366 10 9 km
Aphelion: 5.906 10 9 1 0.2488 7.3754 10 9 km
(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.
4.4977 109
1 0.0086 cos
(c)
7 × 10 9
−5 × 10 9 8 × 10 9
−7 × 10 9

860 Chapter 9 Topics in Analytic Geometry
ep
56. (a) Radius of earth 4000 miles. Choose r
.
1 e cos
Vertices: 126,800, 0 and 4119,
a
c 65,459.5 4119 61,340.5
e c a 61,340.5
65,459.5 0.937
2a
126,800 4119
2
ep
1 e cos 0 ep
1 e cos ep
1 e ep
1 e 2ep
1 e 2
ep
7988.1
Thus, p a1 e2
8525.2. Thus, r
.
e
1 e cos 1 0.937 cos
(b) When
and the distance from the surface of the earth to the satellite is
15,029 4000 11,029 miles.
(c) When and distance 38,370 miles.
60, r 15,029
30, r 42,370
65,459.5
57.
4
r
43
3 3 sin 1 sin
False. The directrix is below the pole.
58.
16
r 2
9 4 cos
4
False. The graph is not an ellipse.
(It is two ellipses.)
59.
x 2
a 2 y 2
x 2
a 2 y2
b 2 1
b 1 60. 2
r 2 cos 2
r 2 sin 2
r 2 cos 2
1
r2 sin 2
1
a 2 b 2 a 2 b 2
r 2 cos 2
r 2 1 cos 2
r 2 cos 2
1
r2 1 cos 2
1
a 2 b 2 a 2 b 2
r 2 b 2 cos 2 r 2 a 2 r 2 a 2 cos 2 a 2 b 2
r 2 b 2 cos 2 r 2 a 2 r 2 a 2 cos 2 a 2 b 2
r 2 b 2 a 2 cos 2 r 2 a 2 a 2 b 2
r 2 b 2 a 2 cos 2 r 2 a 2 a 2 b 2
For an ellipse, b 2 a 2 c 2 . Hence,
a 2 b 2 c 2
r 2 c 2 cos 2 r 2 a 2 a 2 b 2
r 2 c 2 cos 2 r 2 a 2 a 2 b 2
r a c 2
2 cos 2 r 2 b 2 , e c a
r 2 e 2 cos 2 r 2 b 2
r 2 e 2 cos 2 r 2 b 2
r 2 1 e 2 cos 2 b 2
r 2 e 2 cos 2 1 b 2
b 2
r
r 2
.
2
1 e 2 cos 2
r 2 c a 2 cos 2 r 2 b 2 , e c a
b 2
e 2 cos 2 1
b 2
1 e 2 cos 2
© Houghton Mifflin Company. All rights reserved.

Section 9.8 Polar Equations of Conics 861
x 2
61.
169 y2
144 1
62.
a 13, b 12, c 5, e 5 13
144
24,336
r 2
1 25169 cos 2 169 25 cos 2
x 2
63.
25 y2
16 1
a 5, b 4, c 3, e 3 5
b 2
16
400
r 2
1 e 2 cos 2 1 925 cos 2 25 9 cos 2
x 2
64.
36 y2
4 1
a 6, b 2, c 40 2 10, e 210
6
r 2
b 2
1 e 2 cos 2
4
1 109 cos 2
10
3
36
10 cos 2 9
x 2
9 y2
16 1
a 3, b 4, c 5, e 5 3
r 2
16
144
1 259 cos 2 25 cos 2 9
65. Center: x, y 0, 0,
c 5, a 4, e 5 4
b 2 c 2 a 2 25 16 9
r 2
b 2
1 e 2 cos 2
⇒ b 3
9
1 2516 cos 2
144
25 cos 2 16
66. Center: x, y 0, 0,
c 4, a 5, e 4 5
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67.
b 2 a 2 c 2 25 16 9
r 2
b 2
1 e 2 cos 2
4
r
1 0.4 cos
Vertical directrix to left of pole
(a) e 0.4 ⇒ ellipse
−9
6
−6
⇒ b 3
9
1 1625 cos 2
9
−9
7
−5
225
25 16 cos 2
9
(b)
4
r
1 0.4 cos
Vertical directrix to right of pole
Graph is reflected in line
r
4
1 0.4 sin
2.
Horizontal directrix below pole
90 rotation counterclockwise

862 Chapter 9 Topics in Analytic Geometry
68. The lengths of the major and
minor axes increase as p
increases.
Example:
r
r
0.52
1 0.5 sin
0.54
1 0.5 sin
69. Answers will vary. 70.
r a sin b cos
r 2 a r sin b r cos
x 2 y 2 ay bx
Circle
71.
43 tan 3 172.
6 cos x 2 1 73.
12 sin 2 9
tan 1 3 3
3
cos x 1 2
sin 2 3 4
6 n
x
3
2n,
5
3 2n
sin ± 3
2
3
n,
2
3 n
74.
9 csc 2 x 10 2 75.
2 cot x 5 cos
2
76.
2 sec 2 csc
4
csc 2 x 4 3
2 cot x 0
2 sec 22
cot x 0
sec 2
x
3
sin 2 x 3 4
sin x ±3
2
n,
2
3 n
x
2 n
cos 1 2
3
2n,
5
3 2n
For Exercises 77–80:
sin u 3 cos u 4 cos v 1 sin v 1 5 , 5 , 2 , 2
77.
79.
cosu v cos u cos v sin u sin v 78.
4 5 1 2 3 5 1 2
1
52
2
10
sinu v sin u cos v sin v cos u 80.
3 5 1 2 1 2 4 5
1
52
2
10
sinu v sin u cos v sin v cos u
3
5 2 1 2 1 4 5
7
52 72
10
cosu v cos u cos v sin u sin v
4
52 1 3
5 2
1
7
52 72
10
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81. 12 C 9 220 82. 18 C 16 153 83. 10 P 3 720
84. 29 P 2 812

Review Exercises for Chapter 9 863
Review Exercises for Chapter 9
1.
Radius 3 0 2 4 0 2 2.
x 2 y 2 25
9 16 25 5
Radius 8 0 2 15 0 2
x 2 y 2 289
64 225 289 17
3.
Radius 1 2 5 12 6 2 2 4.
Radius 1 2 6 22 5 3 2
1 2 36 16 1 52 13
2
Center
5 1
, 6 2
2 2 2, 4
x 2 2 y 4 2 13
1 64 64 42
2
Center
2 6
, 3 5
2 2 2, 1
x 2 2 y 1 2 32
5.
1
2 x2 1 2 y2 18 6.
x 2 y 2 36
Center: 0, 0
Radius: 6
3
4 x2 3 4 y2 1
x 2 y 2 4 3
Center:
Radius:
0, 0
2
3 23
3
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7.
8.
16x 2 x 1 4 16 y 2 3 2 y 9 16 3 4 9
Center:
16x 2 16y 2 16x 24y 3 0
16x 1 2 2 16 y 3 4 2 16
x 1 2 2 y 3 4 2 1
1 2 , 3 4
Radius: 1
4x 2 4y 2 32x 24y 51 0
4 x 2 8x 16 4 y 2 6y 9 51 64 36
4x 4 2 4 y 3 2 49
x 4 2 y 3 2 49 4
Center: 4, 3
Radius: 7 2

864 Chapter 9 Topics In Analytic Geometry
9.
x 2 4x 4 y 2 6y 9 3 4 9 10.
Center:
Radius: 4
x 2 2 y 3 2 16
2, 3
−7 −6 −4 −3 −2 −1 2 3
−2
−3
−4
−5
−6
2
−8
y
x
x 2 8x 16 y 2 10y 25 8 16 25
Center:
Radius: 7
4, 5
x 4 2 y 5 2 49
14
10
8
6
4
2
−14 −10
2 4 6
−4
−6
y
x
11. x-intercepts:
x 3 2 0 1 2 7
12. x-intercepts:
3 ± 6, 0
y-intercepts:
0 3 2 y 1 2 7
No
y-intercepts
x 3 2 6
x 3 ±6
x 3 ± 6
y 1 2 2, impossible
y-intercepts:
x 5 2 0 6 2 27
x 5 2 9, impossible
No x-intercepts
0 5 2 y 6 2 27
y 6 2 2
y 6 ±2
y 6 ± 2
0, 6 ± 2
13.
4x y 2 0 y
14.
Vertex:
Focus:
y 2 41x, p 1
0, 0
1, 0
Directrix: x 1
6
4
2
–2 2 4 6 8 10
–2
–4
–6
x
y 1 8 x2
x 2 42y, p 2
Vertex:
Focus:
0, 0
0, 2
Directrix: y 2
y
3
–12 –9 –6 6 9 12
–6
–9
–12
–15
–18
–21
x
15.
1
2 y2 18x 0 16.
Vertex:
Focus:
1
2 y2 18x
y 2 36x 49x, p 9
0, 0
9, 0
Directrix: x 9
12
−20 −16 −12 −8 −4 4
−4
4
y
x
1
4 y 8x2 0
Vertex:
Focus:
8x 2 1 4 y
x 2 1 32 y 4 1
128y, p 1
128
0, 0
0, 128
1
Directrix: y 1
128
1
8
y
5
16
1
4
3
16
− 3 1 1
16
− 8
− (0, 0)
16
)
1
0,
128)
1
8
3
16
x
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Review Exercises for Chapter 9 865
17. Vertex: 0, 0 18. Vertex: 4, 2
Focus: 6, 0
Focus: 4, 0
Parabola opens to left.
y 2 4px
Vertical axis, p 2
x 4 2 42y 2
y 2 46x
x 4 2 8y 2
y 2 24x
19. Vertex: 6, 4
Passes through 0, 0
Vertical axis
x 6 2 4py 4
0 6 2 4p0 4
36 16p
9 4 p
x 6 2 4 9 4y 4
x 6 2 9y 4
20. Vertex: 0, 5
y 5 2 4px 0 4px
6, 0 on graph:
0 5 2 4p6 ⇒ p 25
24
y 5 2 4 25
24x 25 6 x
21.
x 2 2y 4 1 2 y, p 1 2
22.
2x y 2
Focus:
0, 1 2
4 1 2 x y2
© Houghton Mifflin Company. All rights reserved.
d 1 1 2 b
d 2 2 0 2 2 1 2 2 5 2
d 1 d 2
⇒
Slope of tangent line:
Equation: y 2 2x 2
x-intercept:
1, 0
y
(0, b)
1
2 b 5 2
y 2x 2
F
x
−2 2
−1
d 1
d 2 (2, −2)
−2
⇒
b 2
b 2
0 2 4
2 2
Focus:
p 1 2
1 2 , 0
Let b, 0 be the x-intercept of the tangent line.
d 1 1 2 b
d 2 8 1 2 2 4 0 2 17 2
1
2 b 17
2 ⇒ b 8
m 4 0
8 8 1 4
y 0 1 x 8
4
y 1 4 x 2
x-intercept: 8, 0

866 Chapter 9 Topics In Analytic Geometry
23.
x 2 4py 12
4, 10 on curve:
16 4p10 12 8p ⇒ p 2
x 2 42y 12 8y 96
y x2 96
8
y 0
if x 2 96 ⇒ x 46 ⇒ width is 86 meters.
24. (a) Parabola:
Vertex: 0, 4
Passes through ±4, 0
25.
x 2 4py 4
16 4p0 4
16 16p
1 p
x 2 4y 4
Circle:
Passes through ±4, 0
Radius:
Center:
r 8
0, k
x 2 y k 2 8 2
±4 2 0 k 2 8 2
16 k 2 64
k 2 48
x 2 y 43 2 64
0, ±23
Eccentricity c a
23
4
3
2
k 48 43
3
2
1
−4 −3 −1 1 3 4
–2
–3
(b) Parabola:
x
x 2 4y 4 ⇒ y 1 4 x2 4
Circle:
x 2 y 43 2 64 ⇒ y 64 x 2 43
d 1 4 x2 4 64 x 2 43
d 1 4 x2 64 x 2 4 43
x 0 1 2 3 4
d 2.928 2.741 2.182 1.262 0
x 2
4 y2
16 1 26. x 2
9 y2
8 1
a 4, b 2, c 16 4 12 23
a 3, b 22, c 9 8 1
Center: 0, 0
Center: 0, 0
Vertices: 0, ±4
y
Vertices: ±3, 0
4
Foci:
Foci: ±1, 0
Eccentricity c a 1 3
2
1
−4 −2 −1
−1
1 2 4
−2
−4
y
x
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Review Exercises for Chapter 9 867
27.
x 4 2
6
y 42
9
1 28.
x 1 2
16
y 32
6
1
a 3, b 6, c 9 6 3
a 4, b 6, c 16 6 10
Center:
Vertices:
Foci:
4, 4
4, 1, 4, 7
4, 4 ± 3
Eccentricity c a 3
3
1
y
−2 −1 1 2 3 5 6 7 8
−2
−3
−4
−5
−6
−7
−8
−9
x
Center: 1, 3
Vertices: 3, 3, 5, 3
Foci: 1 ± 10, 3
Eccentricity c a 10
4
9
8
7
6
5
4
2
1
−6 −5 −4 −3 −2 −1 1 2 3 4
y
x
29. (a) 16x 2 2x 1 9y 2 8y 16 16 16 144 (c)
16x 1 2 9y 4 2 144
y
−3 −2 −1 1 2 3 4 5
x
(b) Center: 1, 4
x 1 2
9
y 42
16
1
−2
−3
−4
−5
−6
a 4, b 3, c 16 9 7
−8
Vertices: 1, 0, 1, 8
Foci:
1, 4 ± 7
e c a 7
4
30. (a) 4x 2 4x 4 25y 2 6y 9 141 16 225
(c)
y
© Houghton Mifflin Company. All rights reserved.
31. (a)
4x 2 2 25y 3 2 100
x 2 2
25
(b) a 5, b 2, c 21
Center: 2, 3
Vertices: 3, 3, 7, 3
Foci: 2 ± 21, 3
e 21
5
y 32
4
1
3x 2 4x 4 8y 2 14y 49 403 12 392
—CONTINUED—
3x 2 2 8y 7 2 1
x 2 2
13
y 72
18
1
6
4
2
−8 −6 −4 −2 2 4
−2
−4
−6
x

868 Chapter 9 Topics In Analytic Geometry
31. —CONTINUED—
(b) Center: 2, 7
(c)
y
a 3
3 , b 2
4
8
6
c 2 a 2 b 2 1 3 1 8 5 24 ⇒ c 30
12
3
Vertices:
2 ±
3 , 7
30
Foci:
2 ±
12 , 7
−4
−3
−2
−1
4
2
x
Eccentricity:
c
a 3012
33
10
4
32. (a)
x 2 20y 2 5x 120y 185 0
(b)
a 5
2 , b 1 4 , c 5 4 1 16 19
4
(c)
x2 5x 25 4 20y2 6y 9 185 25 4 180
1
y
x 52 2
54
−1 1 2 3 4
−1
x 5 2 2 20y 3 2 5 4
x
y 32
116
1
Center:
5 2 , 3
Vertices: 5 2 ± 5
2 , 3
Foci: 5 2 ± 19
4 , 3
e c a 194
52 95
10
−2
−3
−4
33. Vertices:
Foci:
±4, 0
±5, 0
a 5, c 4 ⇒ b 3
x 2
25 y2
9 1
34. Vertices: 0, ±6
Passes through 2, 2
Vertical major axis
Center: 0, 0, a 6
x 2
b 2 y2
36 1
2 2
b 2 22
36 1
4
b 2 1 1 9 8 9
b 2 36 8 9 2
x 2
92 y2
36 1
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Review Exercises for Chapter 9 869
35. Vertices: 3, 0, 7, 0
Foci: 0, 0, 4, 0
Horizontal major axis
Center: 2, 0
a 5, c 2,
b 25 4 21
x h 2
a 2
x 2 2
25
y k2
b 2 1
y2
21 1
36. Vertices: 2, 0, 2, 4
Foci: 2, 1, 2, 3
Vertical major axis
Center: 2, 2
a 2, c 1,
b 4 1 3
x h 2
b 2
y k2
a 2 1
x 2 2 y 2
2
1
3 4
37. a 5, b 4, c a 2 b 2 25 16 3 38.
x 2
324 y 2
1, a 324 18, b 196 14
196
The foci should be placed 3 feet on either side of
the center and have the same height as the pillars.
Longest distance: 2a 218 36 feet
Shortest distance: 2b 214 28 feet
c 2 a 2 b 2 128
Foci: ±82, 0
Distance between foci:
162 22.63 feet
39.
a c 1.3495 10 9 40.
a c 1.5045 10 9
Adding,
Then
c 1.5045 10 9 1.427 10 9 0.0775 10 9
e c a 0.0543.
2a 2.854 10 9 ⇒ a 1.427 10 9 .
a 72 2 36
e c 0.2056 ⇒ c ae 7.4016
a
b 2 a 2 c 2 36 2 7.4016 2 1241.2
x 2
1296 y2
1241.2 1
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41. (a)
(b)
5y 2 4x 2 20
y 2
4 x2
5 1
a 2, b 5,
c 4 5 3
Center:
Vertices:
Foci:
0, 0
0, ±3
0, ±2
Eccentricity c a 3 2
(c)
5
4
3
1
−5 −4 −3 −2 −1 1 2 3 4 5
−3
−4
−5
y
x

870 Chapter 9 Topics In Analytic Geometry
42. (a)
x 2 y 2 9 4
x 2
94
y2
94 1
(b)
a 3 2 , b 3 2 , (c)
c
9
4 9 4
−4 −3 −1
9
2 3 2 32
2
Center:
Vertices:
Foci:
0, 0
±3 2 , 0
±32 2 , 0
5
4
3
2
1
−5 1 3 4 5
−2
−3
−4
−5
y
x
Eccentricity c a 322
32
2
43. (a) 9x 2 2x 1 16y 2 2y 1 151 9 16 (c)
y
9x 1 2 16y 1 2 144
6
4
x 1 2
16
(b) Center: 1, 1, a 4, b 3, c 5
Vertices: 5, 1, 3, 1
y 12
9
1
2
−6 −4 2 4 6
−2
−4
−6
−8
8
x
Foci: 6, 1, 4, 1
Eccentricity: 5 4
44. (a) 25y 2 6y 9 4x 2 2x 1 121 225 4 (c)
y
25y 3 2 4x 1 2 100
y 3 2
4
(b) Center: 1, 3, a 2, b 5, c 29
Vertices: 1, 1, 1, 5
Foci: 1, 3 ± 29
Eccentricity: 29
2
x 12
25
1
6
4
2
−8 −6
−2
4
−4
−6
−8
−10
−12
−14
6 8 10
x
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Review Exercises for Chapter 9 871
45. (a)
y 2 2y 1 4x 2 12x 36 59 1 144
(b) Center:
Foci:
6, 1
y 1 2 4x 6 2 202
x 6 2 y 12
1
1012 202
a 2 101
2 , b2 202, c 2 101 505
202
2 2
Vertices:
2 , 1
6 ± 101
6 ± 505
2 , 1
Eccentricity: e c a 505
101 5
(c)
30
20
10
−30 −20
10 20 30
−10
−20
−30
y
x
46. (a)
9 x 2 8x 16 y 2 8y 16 119 144 16 9
9x 4 2 y 4 2 9
x 4 2
y 42
9
1
(b) Center:
4, 4
(c)
y
a 1, b 3, c 10
Vertices: 4 ± 1, 4: 3, 4, 5, 4
Foci: 4 ± 10, 4
12
10
8
6
4
Eccentricity: e 10
2
−2 4 8 10 12
−2
x
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47.
x 2
a 2 y2
b 2 1
a 4
c 2 a 2 b 2 ⇒ 36 16 b 2
x 2
16 y2
20 1 ⇒ b 20 25
48. Vertices: 0, ±1
Foci: 0, ±3
Vertical transverse axis
Center: 0, 0
a 1, c 3, b 9 1 8
y 2
1 x2
8 1

872 Chapter 9 Topics In Analytic Geometry
49. Foci: 0, 0, 8, 0 ⇒ c 4 50. Vertical transverse axis
Center: 4, 0
Center: 3, 0 ⇒ c 2
Asymptotes:
a
y ±2x 4 ⇒ b b 2 ⇒ a 2b
a 2 ⇒ b 2a
a 2 b 2 c 2
c 2 a 2 b 2
2b 2 b 2 4
16 a 2 2a 2 5a 2 ⇒ a 4 5 ,
x 4 2
165
b 8 5
y2
645 1
y k 2
a 2
5y 2
16
b 2 4 5
a 2 16
5
x h2
b 2 1
5x 32
4
1
51.
d 2 d 1 186,0000.0005
2a 93
a 46.5
c 100
b c 2 a 2
x 2
a 2 y2
b 2 1
(−100, 0)
B
100
80
60
40
20
y
−60 −20
−40
−60
−80
−100
(100, 0)
x
20 A
(60, 0)
x 60 ⇒ y 2 b 2 x2
a 1 2 1002 46.5 2
60 2
46.5 1 5211.57 ⇒ y 72.2
2
72.2 miles north
52. Let the friends be at B and C, you at the origin A. The sound at C is heard 2 seconds after B:
2a CD BD 21100 5280 5 Thus, a 5 12 . 24 , c 2 and 576 . Thus, using miles,
the hyperbola is
x 2
25576 y 2
Now place the center at 1, 0 and determine the second hyperbola.
c 1 and
b 2 1 25
64 39
64
x 1 2
2564
y2
3964 1
2279576 1. b 2 c 2 a 2 2279
2a DB AD 6
1100
5280 ⇒ a 5 8
1.
y
2.
C
−2
−1
1
−1
A
D
1
2
B
x
C
−2
−1
1
−1
A
y
D
1
2
B
x
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Review Exercises for Chapter 9 873
53.
54.
3x 2 2y 2 12x 12y 29 0
3 x 2 4x 4 2y 2 6y 9 29 12 18
3 x 2 2 2y 3 2 1
Ellipse
4x 2 4y 2 4x 8y 11 0
A C 4 ⇒ Circle
55.
5x 2 2x 1 2y 2 2y 1 17 5 2
Hyperbola
5x 2 2y 2 10x 4y 17 0 56.
5x 1 2 2y 1 2 14
y 1 2
7
x 12
145
1
4y 2 5x 3y 7 0
A 0, C 4, AC 0
Parabola
57.
xy 4 0
A 0, B 1, C 0
cot 2 A C
B
0
⇒
x 2
2 x y, y 2
2 x y
4
y′ 5
x′
4
3
2
−3 −2
−2
y
2 3 4 5
x
xy 4
2
2 x y 2
2 x y 4
1
2 x 2 1 2 y 2 4
x 2
8
y 2
8
1
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58.
Hyperbola
x 2 10xy y 2 1 0
A C 1 ⇒ cot 2 0 ⇒
x 2
2 x y , y 2
2 x y
2
2 x y
2
10
2
2 x y
2
2 x y
2
2 x y
2
1 0
Hyperbola
4
1
2 x 2 1 2 y 2 xy 5x 2 y 2 1 2 x 2 xy 1 2 y 2 1
4x 2 6y 2 1
x 2
14 y 2
16 1
y′
2
1
y
1 2
x′
x

874 Chapter 9 Topics In Analytic Geometry
59.
5x 2 2xy 5y 2 12 0
y
A 5, B 2, C 5
cot 2 0
⇒
x 2
2 x y, y 2
2 x y
4
5x 2 2xy 5y 2 12
y′ 3
2
x′
1
−3 −2 −1 1 2 3
−1
−2
−3
x
5
2
2 x y
2
2
2
2 x y
2
2 x y 5
2
2 x y
2
12
5
1
2 x 2 xy 1 2 y 2 x 2 y 2 5
1
2 x 2 xy 1 2 y 2 12
4 x 2 6y 2 12
x 2
3
y 2
2
1
Ellipse
60.
cot 2 4 4
8
x 2
2 x y, y 2
2 x y
4 2
2 x y
2
8 2
2 x y 2
2 x y
4 2
2 x y
2
2 x 2 y 2 2xy 4x 2 y 2 2x 2 y 2 2xy 7x y 9x y 0
Parabola
(c)
0 ⇒
−5 2
2
−10
4
72 2
2 x y
92 2
2 x y
0
y′
−4 −3
2 3 4
−4
8x 2 16x 2y 0
−6 6
−4
4
3
2
y
y 4x 2 8x
61. (a) B 2 4AC 8 2 4161 0 62. (a) B 2 4AC 300 ⇒ Ellipse
Parabola
(b) 7y 2 8xy 13x 2 45 0
(b) y 2 5 8xy 16x 2 10x 0
y 8x ± 64x2 4713x 2 45
y 8x 5 ± 5 8x2 416x 2 14
10x
2
(c)
4
x′
x
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Review Exercises for Chapter 9 875
63. (a) B 2 4AC 2 2 411 0
(c)
Parabola
(b) y 2 2x 22y x 2 22x 2 0
y 22 2x ± 2x 222 4x 2 22x 2
2
10
−15 0
0
64. (a)
(b)
(c)
B 2 4AC 100 4
y 2 10xy x 2 1 0
y 10x ± 100x2 4x 2 1
2
y 10x ± 96x2 4
2
y 5x ± 24x 2 1
4
96 > 0 ⇒ Hyperbola
65. Adding the equations,
24x 240 0 ⇒ x 10. Then:
4 100 y 2 560 24y 304 0
y 2 24y 144 0
y 12 2 0 ⇒ y 12
Solution: 10, 12
−6 6
−4
66.
4x 2 4y 2 100
9x 4y 2 0
Adding:
4x 2 9x 100
4x 2 9x 100 0
67.
t 2 1 0 1 2 3
x 8 5 2 1 4 7
y 15 11 7 3 1 5
16
y
x 44x 25 0 ⇒ x 4, 25 4
12
If x 4:
8
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68.
4y 2 94 36
y 2 9 ⇒ y ±3
If x 25 4 , 925 4 4y 2 , impossible.
Answer: 4, 3, 4, 3
x t, y 8 t
t 0 1 2 3 4
x 0 1 2 3 2
y 8 7 6 5 4
y
8
7
6
5
4
3
2
1
−3 −2 −1 1 2 4 5 6
−1
x
4
−12 −8 −4
8
−4
x

876 Chapter 9 Topics In Analytic Geometry
69.
x 5t 1, y 2t 5 70.
t 1 5x 1 ⇒
y 2 5x 1 5 2 5 x 27 5 , line
y
40
30
x 4t 1, y 8 3t
t 1 4x 1 ⇒
y 8 3 1 4x 1 3 4 x 35 4
y
20
15
20
10
−10
−10
−20
10 20
30
x
5
−20 −15 −10 −5
−10
−15
−20
5
10 20
x
71.
x t 2 2, y 4t 2 3 72.
x ln 4t, y t 2
t 2 x 2 ⇒
e x 4t, t 1 4 ex
⇒
y 1 4 ex 2 1 16 e2x
y 4x 2 3 4x 11, x ≥ 2
y
y
4
3
12
2
10
8
6
1
−4 −3 −2 −1 1 2 3 4
x
4
−2
2
−3
−4 −2 4 6 8 10 12
x
−4
−4
73.
75.
y
x t 3 , y 1 2 t2 74.
t x 13 ⇒ y 1 2 x23
−4 −3 −2 −1 1
−1
−2
−3
x 3 t 76.
y t
t x 3 ⇒ y t x 3
y x 3
4
−6 6
−4
5
4
3
2
1
2 3 4
x
x 4 t , y t 2 1
t 4
x
⇒
y 16
x 2 1
x t 77.
y 3 t 3 x x 13
2
−3 3
−2
x 1 t
y t
y t 1 x
20
y
−12 −8 −4 4 8 12
−4
4
−6 6
−4
x
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Review Exercises for Chapter 9 877
78.
x t 79.
x 2t 80.
x t 2 , y t
y 1 t 1 x
y 4t
y 22t 2x
t y 2
x y 2 2 y 4 , y ≥ 0
4
Line
y 4 x, x ≥ 0
−6 6
4
3
−4
−6 6
−1 5
−4
−1
81.
x 1 4t
y 2 3t
t x 1
4
⇒ y 2 3
x 1
4 2 3 4 x 3 4
3x 4y 11 0
y 11
4 3 4 x −4 8
−2
6
82.
x t 4, y t 2 7
83.
t x 4
y x 4 2
−2 10
−1
x 3
y t
Vertical line: x 3
4
−3 9
−4
84.
x t
7
85.
x 6 cos , y 6 sin
8
y 2
cos x 6 , sin y 6
−12
12
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86.
x 3 3 cos , y 2 5 sin
cos x 3
3 , sin y 2
5
x 3 2
9
y 22
25
1
−6 6
−4
8
−4
−1
14
87.
x 2
36 y2
36 1
x 2 y 2 36
y 6x 2
x t, y 6t 2
x t, y 6t 2
Other answers possible
−8

878 Chapter 9 Topics In Analytic Geometry
88.
x t, y 10 t 89.
x t, y 10 t
Many answers possible
y x 2 2
x t, y t 2 2
x t 1, y t 1 2 2 t 2 2t 3
Other answers possible
90.
x t, y 2t 3 5t 91.
x t, y 2t 3 5t
Many answers possible
x x 1 tx 2 x 1 3 t8 3 5t 3
y y 1 ty 2 y 1 5 t0 0 5
or x t, y 5
92.
x x 1 tx 2 x 1 2 t2 2 2 93.
y y 1 ty 2 y 1
1 t4 1 1 5t
or x 2, y t
x x 1 tx 2 x 1
1 t10 1 11t 1
y y 1 ty 2 y 1 6 t0 6 6t 6
94.
x x 1 tx 2 x 1 0 t 5 2 0 5 2
y y 1 ty 2 y 1 0 t6 0 6t
or x 5t, y 12t
t 95. 90, 4 is on the curve:
90 0.82v 0 t ⇒ v 0 90
0.82t
4 7 0.57
90
0.82t t 16t 2 ⇒
16t 2 3 0.5790
0.82
Hence, v 0
⇒ t 2.024
90
54.23 ftsec.
0.822.024
96. From Exercise 95, v 0 54.23. 97. From Exercise 96:
99.
x 0.8254.23t 44.47t
y 7 0.5754.23t 16t 2
7 30.91t 16t 2
π
2
( 1, 4
π
1 2 3
1, 7
4 , 5
1, 4 , 1, 3
4
(
0
25
0 100
0
The maximum height is approximately
21.9 feet for t 0.97.
1,
100.
4
(
−5, −
π
3
5,
(
5
π
2
3 , 5,
98. From Exercise 95,
t 2.024 seconds.
1 2 3 4 5
2
3 , 0
5, 4
3
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)
Review Exercises for Chapter 9 879
101.
r, 2, 11
6 102.
r, 1, 5
6
π
2
5
( π
1,
6 )
π
2
11 π
−2, −
6 )
1 2 3
0
1 2 3
0
2,
6 ,
2,
7
6 ,
2, 5
6
1,
6 ,
1,
11
6 ,
1, 7
6
103.
5, 4
3 104.
( 10,
3π
4 )
π
2
π
2
(
5, −
4
3
π
(
1 2 3 4
0
1 2 3
0
2
5,
3 , 5, 3 ,
5
5,
3
r, 10, 5
4 , 10, 4 ,
7
10,
4
105.
r, 5, 7
6 106.
x, y 53
2 , 2
5
π
2
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107.
(
5, −
7π
6
2, 5
3
(
π
2
1 2 3
x r cos 2
1
y r sin 2
3
2 3
x, y 1, 3
0
2 1 3
0
π
2
)
5 π
2, − )
1 2 3 4
1 2 3 4 5
−4,
2π
3
r, 4, 2
3
(
(
0
x, y 4 cos 2 2
, 4 sin
3 3 2, 23

880 Chapter 9 Topics In Analytic Geometry
108.
11
1, 6 109.
x r cos 1
3
2
y r sin 1 1 2
x, y 3
2 , 1 2
)
11 π
−1,
6 )
π
2
1 2 3
0
r, 3, 3
4
x, y 3 cos 3 3
, 3 sin
4
(
3,
3π
4
(
π
2
1 2 3
0
4 32
2
, 32
2
110. r, 0, the origin
x, y 0, 0
π
2
x, y 0, 9
r, 9, 3
2 , 9,
y
2
2 , 111. −3 3 6 9 12
3
( 0, 2
π
(
1 2 3 4 5
0
−3
−6
x
−9
(0, −9)
−12
112.
114.
y
x, y 3, 4 113.
r 5, tan 4
3
5, 126.87, 5, 306.87
or
5, 2.214, 5, 5.356
(radians)
x, y 3, 3
Third quadrant,
7
r 2 3 2 3 12
r, 23, 7
6 , 23,
6
⇒
(−3, 4)
4
3
2
1
−4 −3 −2 −1 1 2 3 4
−1
6
−2
−3
−4
r 23
x
−4 −3 −2 −1 1 2 3 4
−1
(−3, − 3
(
4
3
2
1
−2
−3
−4
y
x, y 5, 5
r, 52, 7
4 ,
2
1
y
−2 −1
−1
1 2 3 4 5 6
−2
−3
−4
−5 (5, −5)
−6
x
3
52, 4
x
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Review Exercises for Chapter 9 881
115.
x 2 y 2 9 116.
x 2 y 2 20 117.
x 2 y 2 4x 0 118.
x 2 y 2 6y
r 2 9
r 2 20
r 2 4r cos 0
r 2 6r sin
r 3
r 25
r 4 cos
r 6 sin
119.
r cos r sin 5
xy 5 120.
r 2 5 csc sec
xy 2
r cos r sin 2
r 2 2 sec csc
121.
4 r cos 2 r sin 2 1
4r 2 cos 2 r 2 1 cos 2 1
4x 2 y 1 122.
r 2 3 cos 2 1 1
r 2
1
3 cos 2 1
2x 2 3y 2 1
2 r cos 2 3r sin 2 1
r 2 2 cos 2 3 sin 2 1
r 2 21 sin 2 3 sin 2 1
r 2 2 sin 2 1
r 2
1
2 sin 2
123.
r 5 124.
x 2 y 2 5 2 25
r 12 125.
x 2 y 2 144
r 3 cos
r 2 3r cos
Circle
Circle
x 2 y 2 3x
126.
r 8 sin
r 2 8r sin
x 2 y 2 8y
127.
r 2 cos 2
r 2 1 2 sin 2
r 4 r 2 2r 2 sin 2
x 2 y 2 2 x 2 y 2 2y 2
x 2 y 2 2 x 2 y 2 0
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128.
r 2 sin
r 3 r sin
x 2 y 2 32 y or
x 2 y 2 3 y 2
129.
5
6
tan y x 1 3
y 3
3 x,
line
130.
4
3
tan y x 3
y 3x
131. r 5, circle
132. r 3, circle
133. y-axis
2 ,
π
2
2
4 6
0
π
2
1 2 4
0
π
2
1
2 3
0

882 Chapter 9 Topics In Analytic Geometry
5
134. line
r 5 cos ,136. r 2 sin , circle
6 , 135. circle 0
π
2
π
2
π
2
1 2 3 4
0
1
2
3
4
6
1 2 3
0
137.
r 5 4 cos
Dimpled limaçon
π
2
Symmetric with respect to polar axis
is maximum at
r 0
(No zeros)
0:
r 0
2 4 6 8 10 12
r, 9, 0
138.
r 1 4 sin
Limaçon with inner loop
π
2
Symmetric with respect to
2
r
is a maximum at
2 : 5,
2
1 2 3 4
0
r 0 when 4 sin 1
⇒
sin 1 4
⇒
3.394, 6.031
139.
140.
r 3 5 sin
Limaçon with loop
Symmetry: line
r -
r 8
Maximum value: when
2
0.6435, 2.4981
3
Zeros: r 0 when sin 3 5
r 2 6 cos
Limaçon with inner loop
Symmetry: polar axis
Maximum: when
Zero: r 0 when cos 1 3 ⇒
2
1.231, 5.052
π
2
1 2 3 5
0
r 8 2
π
2
0
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Review Exercises for Chapter 9 883
141.
r 3 cos 2, 0 ≤
Four-leaved rose
Symmetric with respect to polar axis, and pole
2 , 0
The value of is a maximum (3) at
2 , , 3
4
r
2 .
r 0 for
4 , 3
4 , 5
4 , 7
4
≤ 2
0,
π
2
142.
r cos 5
Five-leaved rose
Symmetric with respect to polar axis
r
is maximum value of 1 at
n
r 0 for
10 2n , n 0, 1, 2, . . .
10
, n 0, 1, 2, . . .
5
π
2
1
0
143.
r 2 5 sin 2
Lemniscate
Symmetry with respect to pole
Maximum value: 5 when
Zeros: r 0 when 0,
2 , , 3
2
0
r - 1 2 3
4 , 5
4
π
2
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144.
145.
r 2 cos 2
Lemniscate
Symmetry: Pole, polar axis, and line
r 1
Maximum: when
Zeros: r 0 when
4 , 3
4 , 5
4 , 7
4
r
e 1
Parabola
−6
2
1 sin
6
6
0, , 2
2 0
1
146.
π
2
1
r
, e 2
1 2 sin
Hyperbola symmetric with and having
vertices at 13, 2 and 1, 32
−3
3
3
2
−2
−1

884 Chapter 9 Topics In Analytic Geometry
147.
r
4
5 3 cos
2
148.
r
6
6
1 4 cos 1 4 cos
45
1 35 cos
−3
3
Hyperbola e 4
4
e 3 5
−6
6
−2
Ellipse
−4
149.
r
5
6 2 sin
2
150.
r
3
34
4 4 cos 1 cos
56
1 13 sin
−3
3
Parabola e 1
4
e 1 3
−4
8
−2
151.
Ellipse
e 1
r
4
1 cos
Vertical directrix: x 4
−4
ep
152. Parabola: r , e 1
1 e sin
Vertex: 2,
Focus: 0, 0 ⇒ p 4
r
2
4
1 sin
153. Ellipse:
Vertices:
One focus: 0, 0 ⇒ c 2
23 p
e c 5
1 23 cos 0 ⇒ p 5 a 2 3 , 2
r
r
ep
1 e cos
5, 0, 1, ⇒ a 3
2352
1 23 cos
53
1 23 cos
ep
154. Hyperbola: r
1 e cos
Vertices: 1, 0, 7, 0 ⇒ a 3
One focus: 0, 0 ⇒ c 4
43 p
e c 1
1 43 cos 0 ⇒ p 7 a 4 3 , 4
5
3 2 cos
73
7
r
4374
1 43 cos 1 43 cos 3 4 cos
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Review Exercises for Chapter 9 885
155.
e 0.093
ep
Use r
.
1 e cos
2a
r
1.512
1 0.093 cos
Perihelion:
Aphelion:
0.093p
1 0.093 cos 0 0.093p
0.1876p 3.05 ⇒ p 16.258, ep 1.512
1 0.093 cos
1.512
1 0.093 1.383
1.512
1 0.093 1.667
astronomical units
astronomical units
ep
156. Use r
(horizontal directrix below pole).
1 e sin
e 1 (parabola)
When
r
r 12,000,000
1 sin
When
p
1 sin
2 p 2
2 , r 6,000,000.
6,000,000 ⇒ p 12,000,000
distance is approximately 6,430,781 miles.
3 ,
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157. False. The y 4 -term is not
second degree.
160. (a) Major axis horizontal
(b) Circle
(c) Ellipse is flatter.
(d) Horizontal translation
158. False. There are many sets
possible. For example,
x t, y 3 2t
x 3t, y 3 6t.
159. (a) Vertical translation
(b) Horizontal translation
(c) Reflection in the y-axis
(d) Parabola opens more slowly.
161. The number b must be less than 5. The ellipse
becomes more circular and approaches a circle
of radius 5.
162. The orientation of the graph would be reversed. 163. (a) The speed would double.
(b) The elliptical orbit would be flatter. The length
of the major axis is greater.

886 Chapter 9 Topics In Analytic Geometry
Chapter 9
Practice Test
1. Find the vertex, focus and directrix of the parabola x 2 6x 4y 1 0.
2. Find an equation of the parabola with its vertex at 2, 5 and focus at 2, 6.
3. Find the center, foci, vertices, and eccentricity of the ellipse x 2 4y 2 2x 32y 61 0.
4. Find an equation of the ellipse with vertices 0, ±6 and eccentricity e 1 2 .
5. Find the center, vertices, foci, and asymptotes of the hyperbola 16y 2 x 2 6x 128y 231 0.
6. Find an equation of the hyperbola with vertices at ±3, 2 and foci at ±5, 2.
7. Rotate the axes to eliminate the xy-term. Sketch the graph of the resulting equation, showing both sets of axes.
5x 2 2xy 5y 2 10 0
8. Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or hyperbola.
(a) 6x 2 2xy y 2 0
(b) x 2 4xy 4y 2 x y 17 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 e 2t , y e 4t
11. Convert the polar point 2, 34 to rectangular coordinates.
12. Convert the rectangular point 3, 1 to polar coordinates.
13. Convert the rectangular equation 4x 3y 12 to polar form.
14. Convert the polar equation r 5 cos to rectangular form.
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 6, 2 and focus at 0, 0.
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