Section 10.2
Introduction to Conics: Parabolas
913
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.
2. An ellipse is formed when a plane intersects only the top
or bottom half of a double-napped cone but is not
perpendicular to the axis of the cone, not parallel to the
side of the cone, and does not intersect the vertex.
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.
4. A hyperbola is formed when a plane intersects both halves
of a double-napped cone, is parallel to the axis of the
cone, and does not intersect the vertex.
7. x2 8y
Vertex: 0, 0
Opens downward since p
is negative; matches graph (d).
6. x2 2y
5. y2 4x
Vertex: 0, 0
Opens to the left since p
is negative; matches graph (e).
Vertex: 0, 0
p 12 > 0
Opens upward; matches graph (b).
9. ( y 1)2 4(x 3)
8. y2 12x
Vertex: 3, 1
Opens to the right since p
is positive; matches graph (a).
Vertex: 0, 0
p 3 < 0
Opens to the left; matches graph (f).
x2 2y
1
p 2 < 0
Opens downward; matches graph (c).
y
Vertex: 0, 0
4 y ⇒ h 0, k 0, p 1
2
1
2
Focus:
Vertex: (0, 0)
Focus: 0,
Vertex: 3, 1
12. y 2x2 ⇒ x2 4 18y
11. y 12 x2
x2
10. x 32 2 y 1
1
2
Directrix: y 1
8
1
2
3
4
−3
−4
5
12
x
− 4 −3 − 2 − 1
Directrix: y y
0, 18
1
−5
4
−6
3
−7
2
1
x
1
−1
2
3
14. y2 3x ⇒ 434 x
13. y2 6x
y2 4 32 x ⇒ h 0, k 0, p 23
Vertex: 0, 0
Vertex: 0, 0
Focus:
y
3
Focus: 2, 0
34, 0
Directrix: x 34
4
3
3
Directrix: x 2
y
4
2
− 6 − 5 − 4 −3 − 2 − 1
x
x
1
2
2
−2
−3
−4
−4
4
6
8
914
Chapter 10
Topics in Analytic Geometry
16. x y2 0
15. x2 6y 0
x2 6y 4
32
y
⇒ h 0, k 0, p Focus: 0, 32 Focus:
1
−4 −3
3
2
Vertex: 0, 0
2
3
2
y
y2 x 4 14 x
y
Vertex: 0, 0
Directrix: y 32
x
1
−1
3
14,
1
0
−5
−4
−3
−2
−1
1
Directrix: x 4
4
x
1
−1
−2
−2
−3
−3
−4
−5
−6
17. (x 1)2 8( y 2) 0
18. x 5 y 12 0
y 12 4 14 x 5
(x 1)2 4(2)( y 2)
h 1, k 2, p 2
Vertex: 5, 1
y
Vertex: (1, 2)
4
Focus: (1, 4)
2
1
21
Focus: 5 4 , 1 ⇒ 4 , 1
3
1
19
Directrix: x 5 4 4
1
Directrix: y 0
− 3 −2 − 1
y
x
1
2
3
4
5
6
−3
4
−4
2
−6
−4
x
−2
2
−2
−4
3
19. x 2 4 y 2
2
20. x 12 4 y 1 41 y 1
2
y
x 32 2 41 y 2
1
Vertex: 2, 1
8
7
6
5
4
3
h 32, k 2, p 1
3
Vertex: 2, 2
3
Focus: 2, 3
1
1
Focus: 2, 1 1 ⇒ 2, 2
Directrix: y 1 1 0
y
8
1
x
−7 −6 −5 −4 −3 −2 −1
Directrix: y 1
6
1 2 3
−2
4
−6
−4
x
−2
2
4
−2
21.
y 14x2 2x 5
y
4y x2 2x 5
4y 5 1 x2
22.
4x y2 2y 1 1 33 y 12 32
6
2x 1
y 12 41x 8
4
4y 4 (x 1)2
Vertex: 8, 1
2
(x 1)2 4(1)( y 1)
h 1, k 1, p 1
Vertex: 1, 1
Focus: 1, 2
Directrix: y 0
x 14 y2 2y 33
−2
x
2
4
y
Focus: 9, 1
4
Directrix: x 7
2
x
2
−2
−4
−6
4
6
10
Section 10.2
915
24. y2 4y 4x 0
23. y2 6y 8x 25 0
y2 4y 4 4x 4
y2 6y 9 8x 25 9
h 2, k 3, p 2
Vertex: (2, 3)
−6
x
−4
−2
Directrix: x 0
4
Focus: 0, 2
2
−8
6
Vertex: 1, 2
y
− 10
y
y 22 41x 1
( y 3)2 4(2)(x 2)
Focus: (4, 3)
Introduction to Conics: Parabolas
Directrix: x 2
x
−4
2
4
−2
−4
−6
−8
26. x2 2x 8y 9 0
25. x2 4x 6y 2 0
x2 2x 1 8y 9 1
x2 4x 6y 2
x 12 8 y 1 42 y 1
3
Vertex: 1, 1
Focus: 1, 3
−8
10
Directrix: y 1
x2 4x 4 6y 2 4
x 22 6 y 1
x 22 4 32 y 1
h 2, k 1, p 32
Vertex: 2, 1
4
−14
−9
10
1
Focus: 2, 2 5
Directrix: y 2
−12
On a graphing calculator, enter:
1
y1 6x2 4x 2
27. y2 x y 0
y2
1
4
y x y 12 2 4 14 x 14 y2 4x 4 41x 1
−10
h 14, k 12, p 14
Vertex:
Focus:
28. y2 4x 4 0
4
1
4
2
Vertex: 1, 0
8
Focus: 0, 0
−4
−4
Directrix: x 2
14, 12 0, 12 20
−8
1
Directrix: x 2
To use a graphing calculator, enter:
y1 12 14 x
1
1
y2 2 4 x
29. Vertex: 0, 0 ⇒ h 0, k 0
Graph opens upward.
x2 4py
Point on graph: 3, 6
32 4p(6)
9 24p
3
8
p
Thus, x2 438 y ⇒ x2 32 y.
30. Point: 2, 6
31. Vertex: (0, 0) ⇒ h 0, k 0
x
3
3
Focus: 0, 2 ⇒ p 2
ay2
2 a62
1
18
a
1 2
x 18
y
y2 18x
x2 4py
x2 4 32 y
x2 6y
916
Chapter 10
32. Focus:
52, 0
Topics in Analytic Geometry
⇒ p 52
33. Vertex: (0, 0) ⇒ h 0, k 0
y 2 4px
Focus: (2, 0) ⇒ p 2
y 2 10x
y2 4px
34. Focus: 0, 2 ⇒ p 2
x2 4py
x2 8y
y2 4(2)x
y2 8x
35. Vertex: (0, 0) ⇒ h 0, k 0
36. Directrix: y 3 ⇒ p 3
37. Vertex: (0, 0) ⇒ h 0, k 0
Directrix: y 1 ⇒ p 1
x2 4py
Directrix: x 2 ⇒ p 2
x2 4py
x2 12y
y2 4px
x2 41y
y2 42x
x2 4y
y2 8x
38. Directrix: x 3 ⇒ p 3
y2
4px
y2
12x
39. Vertex: (0, 0) ⇒ h 0, k 0
Horizontal axis and passes through
the point (4, 6)
y2
4px
x2 4py
32 4p3
9 12p
62 4p(4)
36 16p ⇒ p 40. Vertical axis
Passes through: 3, 3
9
4
y2 494 x
p 34
x2 3y
y2 9x
41. Vertex: 3, 1 and opens downward.
Passes through 2, 0 and 4, 0.
y (x 2)(x 4)
x2 6x 8
(x 3)2 1
(x 3)2 ( y 1)
42. Vertex: 5, 3 ⇒ h 5, k 3
Passes through: 4.5, 4
y k2 4px h
y 32 4px 5
1 4p4.5 5
p 12
y 32 2x 5
44. Vertex: 3, 3 ⇒ h 3, k 3
Passes through: 0, 0
x h2 4p y k
x 32 4p y 3
0 32 4p0 3
9 12p
p 34
x 32 3 y 3
45. Vertex: 5, 2
Focus: 3, 2
Horizontal axis
p 3 5 2
y 2 42x 5
2
y 22 8x 5
43. Vertex: (4, 0) and opens to
the right.
Passes through 0, 4.
(y 0)2 4p(x 4)
42 4p(0 4)
16 16p
1p
y2 4(x 4)
46. Vertex: 1, 2 ⇒ h 1,
k2
Focus: 1, 0 ⇒ p 2
x h2 4p y k
x 12 42 y 2
x 12 8 y 2
Section 10.2
47. Vertex: 0, 4
Directrix: y 2
Vertical axis
Introduction to Conics: Parabolas
49. Focus: 2, 2
Directrix: x 2
Horizontal axis
Vertex: 0, 2
48. Vertex: 2, 1 ⇒ h 2,
k1
Directrix: x 1 ⇒ p 3
p422
y k2 4px h
x 02 42y 4
p202
y 12 43x 2
x2 8 y 4
( y 2)2 4(2)(x 0)
y 12 12x 2
50. Focus: 0, 0
( y 2)2 8x
51. ( y 3)2 6(x 1)
Directrix: y 8 ⇒ p 4
52. y 12 2x 4
y 1 ± 2x 4
For the upper half of the parabola:
y 3 6(x 1)
⇒ h 0, k 4
y 1 ± 2x 4
y 6(x 1) 3
x h2 4p y k
917
Lower half of parabola:
y 1 2x 4
x2 44 y 4
x2 16 y 4
53. y2 8x 0 ⇒ y ± 8x
1 2
54. x2 12y 0 ⇒ y1 12
x
10
x y 3 0 ⇒ y2 3 x
xy20 ⇒ yx2
−5
25
−10
1
2
x2 4
Point: (4, 8)
12
4
2 y x
1
2
1
b
2
d2 4 0 8 21
Focus:
17
2
d1 1
b
2
d2 3 0 92 21
p
2
d1 d2 ⇒ b 8
Slope: m 8 8
4
40
y 4x 8 ⇒ 0 4x y 8
x-intercept: (2, 0)
−10
12y
d1 2
12
x 2 2y
56.
Focus: 0,
−12
Using the trace or intersect
feature, the point of tangency is
6, 3.
The point of tangency is (2, 4).
55. x2 2y ⇒ p 6
1
2
0, 2
1
2
2
5
1
b5
2
9
2
92 92
3
m
03
9
Tangent line: y 3x ⇒ 6x 2y 9 0
2
3
x-intercept: , 0
2
b
918
Chapter 10
Topics in Analytic Geometry
1
1
57. y 2x2 ⇒ x2 y ⇒ p 2
8
Point: (1, 2)
1
y x2
2
1
8
1
1
d1 b b 8
8
Focus: 0, d2 81y x
4 2
2
Focus:
17
8
d1 d2 ⇒ b 2
2 2
4
1 0
1
8
1
0, 8
d1 1
b
8
d2 2 0 8 81
2
2
65
8
65
1
b
8
8
y 4x 2 ⇒ 0 4x y 2
1
x-intercept: , 0
2
2
p
1 0 2 81
Slope: m y 2x2
58.
b
m
64
8
8
8 8
8
20
Tangent line: y 8x 8 ⇒ 8x y 8 0
x-intercept: 1, 0
x 1062 45R 14,045
59.
x2 212x 11,236 45R
60. Maximum revenue occurs at x 135.
11,236
30,000
R 265x 54x2
The revenue is maximum
when x 106 units.
15,000
0
275
0
0
225
0
61. Vertex: 0, 0 ⇒ h 0, k 0
Focus: 0, 4.5 ⇒ p 4.5
(x h)2 4p(y k)
(x 0)2 4(4.5)(y 0)
1 2
x2 18y or y 18
x
62. (a)
(b) Vertex: 0, 0; opens upward
y
y 0 ax 0
2
(−640, 152)
(640, 152)
152 a6402
152
a
6402
x
19
a
51,200
An equation of the cables is
y
19
x2.
51,200
(c)
Distance, x
Height, y
0
0
250
23.19
400
59.38
500
92.77
1000
371.09
Section 10.2
63. (a) Vertex: 0, 0 ⇒ h 0, k 0
Points on the parabola: ± 16, 0.4
919
(b) When y 0.1 we have
1 2
0.1 640
x
x2 4py
64 x2
± 162 4p0.4
± 8 x.
256 1.6p
Introduction to Conics: Parabolas
Thus, 8 feet away from the center of the road, the
road surface is 0.1 foot lower than in the middle.
160 p
x2 4160y
x2 640y
1
y 640x2
64. Vertex: 0, 0
65. (a) V 17,5002 mihr
y 0 4px 0
2
24,750 mihr
y2 4px
At 1000, 800: 8002 4p1000 ⇒ p 160
y2 4160x
x 02 44100 y 4100
x2 16,400 y 4100
y2 640x
66. (a)
(b) p 4100, h, k 0, 4100
67. (a) x2 12
322
y 75
16
x2 64 y 75
(b) When y 0, x2 6475 4800.
0
18
Thus, x 4800 403 69.3 feet.
0
(b) Highest point: 6.25, 7.125
Range: 15.69 feet
68.
540 mi
1 hr
5280 ft
1 mi
1 hr
60 min 1 min
792 fts
60 s
s 30,000
69. False. It is not possible for a parabola to intersect its
directrix. If the graph crossed the directrix there would
exist points closer to the directrix than the focus.
The crate hits the ground when y 0.
x2 v2
y s
16
x2 7922
0 30,000
16
x2 1,176,120,000
x 34,295
The distance is about 34,295 feet.
70. True. If the axis (line connecting the vertex and focus) is horizontal, then the directrix must be vertical.