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A187
Answers to Odd-Numbered Exercises and Tests
(b)
Chapter 10
Section 10.1
1. inclination
m2 m1
3.
1 m1m2
1.
9.
13.
17.
21.
25.
29.
33.
5
(page 732)
2
2. tan Ax1 By1 C
4.
A2 B2
3. 1
5. 3
1
m
−4 −3 −2 −1
37. 0
45.
(a)
y
6
6
12
5
9
6
2
3
1
x
x
−3
3
6
9
− 4 − 3 − 2 −1
−1
12
−3
1
2
3
4
−2
Section 10.2
(page 740)
Vocabulary Check
1. conic
4. axis
4
B
4
B
3
3
(page 740)
2. locus
5. vertex
3. parabola; directrix; focus
6. focal chord
7. tangent
2
2
A
1
C
(b) 4
4
5
5
−1 A
−1
3
61. x-intercept: 7, 0
63. x-intercepts: 5 ± 5, 0
y-intercept: 0, 49
y-intercept: 0, 20
7 ± 53
65. x-intercepts:
,0
2
y-intercept: 0, 1
2
2
324
67. f x 3x 13 49
69. f x 5x 17
3
5 5
324
Vertex: 13, 49
Vertex: 17
3
5, 5 1 2
289
71. f x 6x 12 24
1
Vertex: 12
, 289
24 y
y
73.
75.
7. 3.2236
837
43.
1.3152
37
47.
y
(a)
41. 7
2
−2
3
3
radians, 135
11. radian, 45
4
4
0.6435 radian, 36.9
15. 1.0517 radians, 60.3
2.1112 radians, 121.0
19. 1.2490 radians, 71.6
2.1112 radians, 121.0
23. 1.1071 radians, 63.4
0.1974 radian, 11.3
27. 1.4289 radians, 81.9
0.9273 radian, 53.1
31. 0.8187 radian, 46.9
2, 1 ↔ 4, 4: slope 32
4, 4 ↔ 6, 2: slope 1
6, 2 ↔ 2, 1: slope 14
2, 1: 42.3; 4, 4: 78.7; 6, 2: 59.0
4, 1 ↔ 3, 2: slope 37
3, 2 ↔ 1, 0: slope 1
1, 0 ↔ 4, 1: slope 15
4, 1: 11.9; 3, 2: 21.8; 1, 0: 146.3
7
39.
5
1
(d) The graph has a horizontal asymptote at d 0.
As the slope becomes
larger, the distance
between the origin and
the line y mx 4,
becomes smaller and
approaches 0.
1
2
3
(c) 8
4
x
5
6
1
−2 − 1
−1
C
1
2
3
x
4
5
−2
(b)
35
3537
(c)
74
8
53. 31.0
49. 22
51. 0.1003, 1054 feet
55. 33.69; 56.31
57. True. The inclination of a line is related to its slope by
m tan . If the angle is greater than 2 but less than ,
then the angle is in the second quadrant, where the tangent
function is negative.
4
59. (a) d m 2 1
1. A circle is formed when a plane intersects the top or
bottom half of a double-napped cone and is perpendicular
to the axis of the cone.
3. A parabola is formed when a plane intersects the top or
bottom half of a double-napped cone, is parallel to the side
of the cone, and does not intersect the vertex.
5. e
6. b
7. d
8. f
9. a
10. c
11. Vertex: 0, 0
13. Vertex: 0, 0
Focus: 0, 12 Focus: 32, 0
1
Directrix: y 2
Directrix: x 32
y
y
4
5
3
4
3
2
− 6 − 5 −4 − 3 −2 − 1
1
x
−1
1
2
3
−3
−4
x
1
2
CHAPTER 10
35.
3
6
(page 732)
Vocabulary Check
(c) m 0
d
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Answers to Odd-Numbered Exercises and Tests
15. Vertex: 0, 0
Focus: 0, 32 Directrix: y 32
17. Vertex: 1, 2
Focus: 1, 4
Directrix: y 0
y
y
− 4 −3
45. y 2 2 8 x 5
47. x 2 8 y 4
2
49. y 2 8x
51. y 6x 1 3
10
53.
2
4
1
3
3
4
1
−2
−3
−10
x
−3 −2 −1
1
2
3
4
5
−4
−3
−5
−4
−6
19. Vertex: 32, 2
Focus: 32, 3
Directrix: y 1
0
6
4
2
1
x
1 2 3
23. Vertex: 2, 3
Focus: 4, 3
Directrix: x 0
4
p=1
−18
−14
−4
10
As p increases, the graph becomes wider.
(b) 0, 1, 0, 2, 0, 3, 0, 4
(c) 4, 8, 12, 16; 4 p
(d) Easy way to determine two additional points on the
graph
x
75. ± 1, ± 2, ± 4
m 1
2p
1
± 2 , ± 1, ± 2, ± 4, ± 8, ± 16
81. 12, 53, ± 2
f x x3 7x 2 17x 15
B 23.67, C 121.33, c 14.89
C 89, a 1.93, b 2.33
A 16.39, B 23.77, C 139.84
B 24.62, C 90.38, a 10.88
−2
−12
−4
18
−3
x
−6
73.
−8
77.
79.
83.
85.
87.
89.
27. Vertex: 14, 12
Focus: 0, 12 Directrix: x 12
4
−10
p=4
4
2
−6
2
225
x 106 units
1 2
63. (a) y 640
(b) 8 feet
x
65. (a) 17,5002 miles per hour 24,750 miles per hour
(b) x 2 16,400 y 4100
67. (a) x2 64 y 75
(b) 69.3 feet
69. False. If the graph crossed the directrix, there would exist
points closer to the directrix than the focus.
71. (a) p = 3 p = 2 21
25. Vertex: 2, 1
Focus: 2, 12 Directrix: x 2
y
−8
x
−2
−2
57. 4x y 2 0; 12, 0
1 2
61. y 18
x
0
y
8
7
6
5
4
3
− 10
2, 4
55. 4x y 8 0; 2, 0
59. 15,000
21. Vertex: 1, 1
Focus: 1, 2
Directrix: y 0
y
−7 −6 −5 −4 −3 −2 −1
25
2
x
1
−1
−5
2
−4
29. x 2 32 y
31. x 2 6y
33. y 2 8x
2
2
35. x 4y
37. y 8x
39. y 2 9x
2
41. x 3 y 1
43. y 2 4x 4
Section 10.3
(page 750)
Vocabulary Check
1. ellipse; foci
3. minor axis
(page 750)
2. major axis; center
4. eccentricity
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Answers to Odd-Numbered Exercises and Tests
1. b
2. c
3. d
7. Ellipse
Center: 0, 0
Vertices: ± 5, 0
Foci: ± 3, 0
3
Eccentricity:
5
4. f
19. Ellipse
21. Circle
Center: 2, 3
Center: 1, 2
Vertices: 2, 6, 2, 0
Radius: 6
y
Foci: 2, 3 ± 5 5
6
Eccentricity:
3
5. a
6. e
9. Circle
Center: 0, 0
Radius: 5
y
6
2
y
4
y
6
−6
−2
2
−6
4
−2
2
4
4
6
2
4
6
2
4
8
−6
−2
2
−6
6
x
−2
−2
−4
x
2
−4
x
−8 −6
6
2
−6
−4
−10
x
−2
2
−2
−6
11. Ellipse
Center: 0, 0
Vertices: 0, ± 3
13. Ellipse
Center: 3, 5
Vertices:
3, 10, 3, 0
Foci: 3, 8, 3, 2
Eccentricity: 35
Foci: 0, ± 2
Eccentricity: 32
y
23. Ellipse
Center: 3, 1
Vertices: 3, 7, 3, 5
Foci: 3, 1 ± 26 6
Eccentricity:
3
− 10
8
4
2
−8
x
− 4 −2
y
y
12
2
8
25. Ellipse
4
x
−4 −3
−1
6
1
1
3
4
2
−2
x
−8 −6 −4
2
−2
4
6
15. Circle
Center: 0, 1
Radius: 23
y
1
−2
x
−1
1
−4
−4
y
5
6
Center: 3, 4
2
2
5
5
Vertices: 9, , 3, 2
2 −4
2 4 6
10
−2
5
Foci: 3 ± 33, 2
−6
3
−8
Eccentricity:
2
27. Circle
29. Ellipse
Center: 1, 1
Center: 2, 1
Radius: 23
Vertices: 73, 1, 53, 1
26
y
Foci: 34
15 , 1, 15 , 1
4
Eccentricity: 5
3
2
−1
−2
x
y
−3
2
17. Ellipse
Center: 2, 4
Vertices:
3, 4, 1, 4
4 ± 3
Foci:
, 4
2
3
Eccentricity:
2
y
−3
−2
−1
1
−1
3
x
−2
−3
−2
x
−1
2
1
−1
1
−3
x
−4
−5
1
2
3
CHAPTER 10
−6
4
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Answers to Odd-Numbered Exercises and Tests
31.
33.
4
−4
−6
67. False. The graph of x24 y4 1 is not an ellipse. The
degree of y is 4, not 2.
y2
x2
69. (a) A a 20 a (b)
1
196 36
(c)
a 8
9
10
11
12
13
2
5
6
−4
−4
Center: 0, 0
Center: Vertices: 0, ± 5 Vertices: 1
Foci: 0, ± 2 Foci: 12 ± 2, 1
x2
y2
x2
y2
y2
x2
1 37.
1 39.
1
4
16
36 32
36 11
2
2
2
2
21x
y
x 2
y 3
1 43.
1
400
25
1
9
x 2 2 y 3 2
1
16
9
x 2 2 y 4 2
y 4 2
x2
49.
1
1
4
1
48
64
x2
y 42
x 22 y 22
53.
1
1
16
12
4
1
x2
y2
1
25 16
x2
y2
y
(a)
1
59. (a)
321.84 20.89
14
(b)
1
2 , 1
1
2 ± 5,
35.
41.
45.
47.
51.
55.
57.
−21
21
(0, 10)
x
−14
(25, 0)
(−25, 0)
(c) Aphelion:
35.29 astronomical units
Perihelion:
0.59 astronomical unit
x2
y2
(b)
1
625 100
(c) Yes
x2
y2
61. (a)
1
0.04 2.56
y
(b)
A
301.6
311.0
314.2
0
20
0
The shape of an ellipse with a maximum area is a
circle. The maximum area is found when a 10
(verified in part c) and therefore b 10, so the equation produces a circle.
71. Geometric
73. Arithmetic
75. 547
77. 340.15
Section 10.4
(page 760)
Vocabulary Check
(page 760)
1. hyperbola; foci
2. branches
3. transverse axis; center
4. asymptotes
5. Ax 2 Cy 2 Dx Ey F 0
1. b
2. c
3. a
5. Center: 0, 0
Vertices: ± 1, 0
Foci: ± 2, 0
Asymptotes: y ± x
4. d
7. Center: 0, 0
Vertices: 0, ± 5
Foci: 0, ± 106 Asymptotes: y ± 59 x
y
y
10
8
6
4
2
2
(c) The bottom half
x
−2
2
−1
x
0.4
0.8
−2
9. Center: 1, 2
Vertices: 3, 2, 1, 2
Foci: 1 ± 5, 2
Asymptotes:
y 2 ± 12 x 1
−2
63.
65.
y
y
4
( 49 , 7 )
2
−4
(
− 49 , − 7
x
−2
)
2
−2
4
(
9
4, −
7
)
(
(− 3 5 5 ,
2
−4
−2
− 3 55, − 2
)
(3 5 5 , 2)
x
2
)
(
−4
4
3 5
,−
5
2
)
285.9
350
1
− 0.8 − 0.4
301.6
a 10, circle
(d)
2
(− 49 , 7 )
311.0
−8 −6
x
6 8 10
−2
−4
−6
− 10
y
3
2
1
x
1
−4
−5
2
3
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Answers to Odd-Numbered Exercises and Tests
11. Center: 2, 6
Vertices:
17
19
2, , 2, 3
3
Foci:
2, 6
±
y
2
13
6
x
−2
2
4
6
−6
− 10
Asymptotes:
2
y 6 ± x 2
3
13. Center: 2, 3
Vertices: 3, 3, 1, 3
Foci: 2 ± 10, 3
Asymptotes:
y 3 ± 3x 2
− 12
− 14
y
2
4
6
−4
−3
−8
−4
Section 10.5
41.
45.
3π
4
−
π
4
π
4
−2
6
3
3π
4
(page 769)
(page 769)
1. rotation of axes
2. Ax 2 C y 2 Dx Ey F 0
3. invariant under rotation
4. discriminant
x
2
−12
12
1. 3, 0
3 2
3 33 1
3.
2
−8
2
π
2
7.
2
5.
2
y x 1
2
2
9. y ±
2
y
4
y′
10
3 2 2, 22 2
y
2
−8
,
x′
y'
x'
2
1
x
−4 −3 −2
−10
−2
y2
x
23.
1
1
4
12
1
25
2
2
17y
17x
x 4 2 y 2
27.
1
1
1024
64
4
12
y 5 2 x 4 2
y 2 4 x 2 2
31.
1
1
16
9
9
9
y 22 x2
x 22 y 22
35.
1
1
4
4
1
1
x 32 y 22
1
9
4
x2
y2
(a)
(b) 2.403 feet
1
1
1693
43. 125 1, 0 14.83, 0
3300, 2750
Circle
47. Hyperbola
49. Hyperbola
−2
x
−1
1
−1
−3
−2
−4
11.
x 32 2 y 2 2 1
16
16
y
8
x′
6
y′
4
x
−4
2
−4
4
6
8
2
CHAPTER 10
39.
−
Vocabulary Check
8
19. Center: 1, 3
Vertices: 1, 3 ± 2 Foci: 1, 3 ± 25 Asymptotes:
y 3 ± 13x 1
37.
x
3π
2
x
−6
33.
π
2
−1
−2
−4
29.
3π
2
−6
Asymptotes: y ±
−2
25.
1
x
8
−
2
21.
3
x
−6 −4 −2
4
x2
4
3
1
y
y2
4
2
2
15. The graph of this
17. Center: 0, 0
equation is two lines
Vertices: ± 3, 0
intersecting at 1, 3.
Foci: ± 5, 0
−4
51. Parabola
53. Ellipse
55. Parabola
57. Ellipse
59. Circle
61. True. For a hyperbola, c2 a2 b2. The larger the ratio of
b to a, the larger the eccentricity of the hyperbola, e ca.
63. Answers will vary.
x 32
65. y 1 3
67. xx 4x 4
1
4
69. 2xx 62
71. 22x 34x 2 6x 9
y
y
73.
75.
333200_10a_AN.qxd
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Page A192
Answers to Odd-Numbered Exercises and Tests
37. (a) Hyperbola
6x ± 36x2 20x2 4x 22
(b) y 10
6
(c)
y
x 2 y 2
3 1
6
2
3
y'
x'
2
x
−3
2
3
−9
9
−3
−6
1
17. x 12 6 y 6 15. y2 x
y
y′
y
x′
6
2
−6
39. (a) Parabola
4x 1 ± 4x 12 16x2 5x 3
(b) y 8
2
(c)
−2
x′
4
x
−4
7
y′
2
2
−2
−4
−4
x
−4
2
41.
6
1
−6
−15
−9
15
4
3
21.
10
y
6
−2
19.
43.
y
4
−4
x
x
−2
2
4
− 4 −3 −2 −1
6
9
1
3
4
−2
−3
−6
−10
45
23.
25.
−6
18
6
−9
27
−6
−4
31.72
33.69
27. e
28. f
29. b
30. a
31. d
32. c
33. (a) Parabola
8x 5 ± 8x 52 416x 2 10x
(b) y 2
1
(c)
47. 8, 12
49. 0, 8, 12, 8
2, 2, 2, 4
53. 1, 3 , 1, 3 55. No solution
0, 4
0, 32 , 3, 0
True. The graph of the equation can be classified by finding the discriminant. For a graph to be a hyperbola, the
discriminant must be greater than zero. If k ≥ 14, then the
discriminant would be less than or equal to zero.
61. Answers will vary.
y
y
63.
65.
45.
51.
57.
59.
26.57
4
−4
−6
6
4
5
3
4
3
1
2
−4 −3 −2 −1
1
−4
2
1
2
−3
35. (a) Ellipse
6x ± 36x2 2812x2 45
(b) y 14
3
(c)
6
1
5
4
−1
−1
3
−2
2
−3
−4
5
−3
6
y
2
7
−3 − 2 − 1
−1
5
4
−4
69.
y
1
−4
3
−3
−2
67.
2
−2
x
−6 −5 −4 −3 −2 −1
−1
x
1
t
1
2
3
4
5
71. Area 45.11 square units
73. Area 48.60 square units
t
1
2
7
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Page A193
A193
Answers to Odd-Numbered Exercises and Tests
Section 10.6
7. (a)
(page 776)
9. (a)
y
Vocabulary Check
y
(page 776)
1. plane curve; parametric; parameter
2. orientation
3. eliminating the parameter
4
3
2
2
1
1
1. (a)
t
0
1
2
3
4
x
0
1
2
3
2
y
3
2
1
0
1
(b)
x
−2 −1
1
2
3
4
5
−3
−2
−1
6
2
3
−2
−2
(b) y x 2 4x 4
(b) y 11. (a)
y
x
1
−1
x 1
x
13. (a)
y
y
4
14
3
−2
−1
4
12
2
10
2
1
8
1
6
x
1
3
−4
4
−1
2
4
x
−2
2
(c) y 3 x 2
(b) y 4
15. (a)
4
6
8 10 12 14
−4
x
3
2
2
y2 x2
1
9
9
(b)
CHAPTER 10
y
1
−2
2
−2
x
−2 −1
17. (a)
y
y
1
−4 −3
x
−1
1
3
4
4
3
3
2
1
−2
1
−3
−4
x
−3 −2 −1
The graph of the rectangular equation shows the entire
parabola rather than just the right half.
The graph of the rectangular equation continues the
graph into the second and third quadrants.
3. (a)
5. (a)
2
3
1
3
4
5
7
−2
−3
(b)
−3
−4
−4
−5
x 42
y 12 1
4
21. (a)
x2
y2
1
16
4
(b)
19. (a)
y
y
1
x
−1
y
6
5
4
y
4
4
3
3
2
1
−7
− 4 −3 − 2 − 1
−2
−3
−4
(b) y 23 x 3
x
1 2 3
−2
x
−1
1
−1
(b) y 16x2
2
2
1
1
−2 −1
−1
2
x
−1
1
−1
(b) y 2
3
4
x
1
2
3
−2
−3
−4
1
x3
(b) y ln x
4
5
6
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Page A194
Answers to Odd-Numbered Exercises and Tests
23. Each curve represents a portion of the line y 2x 1.
Domain
Orientation
(a) , Left to right
(b) 1, 1
Depends on (c) 0, Right to left
(d) 0, Left to right
x h 2 y k 2
25. y y1 mx x1
27.
1
a2
b2
29. x 6t
31. x 3 4 cos y 3t
y 2 4 sin 33. x 4 cos 35. x 4 sec y 7 sin y 3 tan 37. (a) x t, y 3t 2 (b) x t 2, y 3t 4
39. (a) x t, y t 2
(b) x t 2, y t 2 4t 4
2
41. (a) x t, y t 1 (b) x t 2, y t 2 4t 5
1
1
43. (a) x t, y (b) x t 2, y t
t2
45. 34
47. 6
(d)
Maximum height: 136.1 feet
Range: 544.5 feet
200
0
600
0
59. (a) x 146.67 cos t
y 3 146.67 sin t 16t 2
(b) 50
No
0
450
0
(c)
Yes
60
0
500
0
18
0
0
51
−6
0
49.
51.
4
−6
−6
6
6
−4
−4
53. b
Domain: 2, 2
Range: 1, 1
57. (a) 100
0
4
55. d
Domain: , Range: , Maximum height: 90.7 feet
Range: 209.6 feet
(d) 19.3
61. Answers will vary.
63. x a b sin y a b cos 65. True
xt
y t 2 1 ⇒ y x2 1
x 3t
y 9t 2 1 ⇒ y x 2 1
67. Parametric equations are useful when graphing two functions simultaneously on the same coordinate system. For
example, they are useful when tracking the path of an
object so that the position and the time associated with that
position can be determined.
69. 5, 2
71. 1, 2, 1
73. 75
75. 3
y
250
y
0
(b)
Maximum height: 204.2 feet
Range: 471.6 feet
220
105°
θ′
x
x
θ′
0
500
0
(c)
Maximum height: 60.5 feet
Range: 242.0 feet
100
0
300
0
− 2π
3