Download Conic Sections Questions

Conic Section Homework Problems
Part I – Graph and analyze each of the following. Give the following information where
appropriate:
 Equations of major and minor axis (hyperbolas and ellipses)
 Coordinates of the vertices (all)
 Coordinates of the focal points (all)
 Equation of the directrix (parabolas)
 Equation of the axis of symmetry (parabolas)
 Equations of the Asymptotes (hyperbolas)
1. y  3  2 x  1
2. y  2  
2
1
 x  3 2
2
3. x 
1
 y  2 2
4
4. x  5  2 y  5
2
5. x 2  4 x  y  2  0
6. x 2  6 x  2 y  8  0
7. y 2  8 y  x  5  0
8. 2 x 2  8 x  y  4  0
9. y  2 x 2  6 x  14
10. y  3 x 2  12 x  6
11. x  y 2  8 y  1
12. 2 x  y 2  4 y  2
13.
17.
x2 y2

1
9
4
14.
 x  2  2   y  3 2
9
4
1
18.
x2 y2

1
16 25
15.
x  12   y  42
9
16
1
x2
 y2 1
4
16.
x 2  y  3

1
25
16
2
19.
20.
x2 y2

1
9
9
x  12   y  12
36
x2 y2
21.

1
16 9
x2 y2
22.

1
4
9
y2 x2
23.

1
25 16
y2 x2
24.

1
9 16
2
2

y  3

x  2
25.

2

x  1
2
26.
  y  1
2
2

y  1

x  1
27.

y 2  x  1
28.

1
25
16
4
9
1
29. 9 x  2   4 y 2  36
2
16
1
30. x 2  16 y  1  16
2
9
16
1
31. x 2  4 y  1  1
2
2
32. 4 x 2  9 y 2  1
33. 9 x 2  72 x  16 y 2  32 y  16  0
34. x 2  4 x  8 y  28  0
35. x 2  4 y 2  6 x  8 y  9  0
36. 4 y 2  x 2  2 x  16 y  11  0
37. 4 x 2  y 2  8 x  4 y  8  0
38. 4 x 2  y 2  8 x  4 y  4  0
39. 4 x 2  y 2  8 x  2 y  6  0
40. 9 x 2  y 2  18 x  6 y  0
Part II – Use the given information to determine the equation of the given conic.
41. A parabola whose vertex is at the origin and focus is 0, 1 2 
42. A parabola whose vertex is at the origin and directrix is x  1
1
43. A parabola whose directrix is y  3 and whose focal point is 2,1
44. A parabola whose vertex (turning point) is  2,3 and focal point is  2,5
45. A parabola whose vertex (turning point) is  2,3 and focal point is  5,3
46. An ellipse whose focal points are  2,5 and 6,5 , and has a vertex at 8,5
47. An ellipse whose focal points are 1,5 and 1,1 , and has a vertex at 3,2 
48. An ellipse whose horizontal major axis and vertical minor axis lengths are 6 and 4 respectively and
whose center is the origin
49. An ellipse whose vertical major axis and horizontal minor axis lengths are 8 and 2 respectively and
whose center is on the point  2,3
50. An ellipse whose center is the origin, one focus at 3,0 and one vertex at 5,0 
51. An ellipse whose center is at 2,2 , vertex at 7,2  and focus at 4,2
52. An ellipse with vertices at 4,3 and 4,9  , and a focus at 4,8
53. An ellipse with focal points at 5,1 and  1,1 , and the length of the major axis is 8.
54. A hyperbola with center at 0,0  , focus at 3,0 and a vertex at 1,0 
55. A hyperbola with a focus at 0,6  and vertices at both 0,2  and 0,2 
56. A hyperbola with foci at 3,7  and 7,7  , and a vertex at 6,7 
57. A hyperbola with center at  3,1 , focus at  3,6  and a vertex at  3,4 
58. A hyperbola with a focus at  4,0  and vertices at  4,4  and  4,2 
59. A hyperbola with focal points at 0,2  and 0,2  , and has asymptotes at y   x
Part III – Solve each problem
60. A satellite dish is an antenna with a parabolic shape. If the dish is 10 feet across and 4 feet deep at
the center, at what position should a receiver be placed?
61. In a car headlight, the bulb is placed at focus of a parabolic reflector. If the reflector is 4 inches in
diameter and 1 inch deep at its center, how far from the center should the bulb be placed?
62. A parabolic reflector for a telescope is 10 inches in diameter and 1 inch deep at its center. Where
will the light reflected off it be concentrated?
63. A suspension bridge has cables which are
in the shape of a parabola. If the
supporting towers are 600 feet apart and
80 feet tall, how far above the roadway is
the cable at a point 150 feet from the
center of the bridge? Assume the
roadway is flat.
For questions 65 to 68, use the information below:
All planets follow a path around the Sun
that is the shape of an ellipse with the Sun
at its center. The distance of the planet to
the Sun when the planet is closest to the
Sun is called the perihelion. The distance
of the planet to the Sun when it is farthest
from the Sun is called the aphelion. The
distance from the center of the orbit (the
center of the ellipse) to either of the far
ends is called the mean distance.
64. The mean distance of Earth to the Sun is 93 million miles. If the aphelion of Earth is 94.5 million
miles, what is its perihelion? Write an equation for the orbit of Earth around the Sun (treat the center
of the orbit as if it were on the origin of a coordinate system).
65. The mean distance of Mars to the Sun is 142 million miles. If the perihelion of Mars is 128.5 million
miles, what is its aphelion? Write an equation for the orbit of Mars around the Sun (treat the center
of the orbit as if it were on the origin of a coordinate system).
66. The aphelion of Jupiter is 507 million miles. If the distance from the Sun to the center of Jupiter’s
Elliptical orbit is 23.2 million miles, what is the perihelion and what is the mean distance? Write an
equation for the orbit of Jupiter around the Sun (treat the center of the orbit as if it were on the origin
of a coordinate system).
67. Pluto’s Perihelion is 4.551 billion miles and the distance from the Sun to the center of the orbit is
897.5 million miles. Find the aphelion of Pluto, as well as its mean distance. Write an equation for
the orbit of Pluto around the Sun (treat the center of the orbit as if it were on the origin of a
coordinate system).
68. A hyperbolic mirror used in telescopes has the property that a light
ray aimed at its focus is will be reflected to the other focus. If the
focus of the mirror is 48 inches from the receiver and the mirror’s
mount is 24 inches above the focus of the mirror, find how far the
vertex of the mirror is from the receiver (see diagram on right).
69. Some comets follow hyperbolic paths with the Sun being the focus of the orbit and the point where
the comet is closest to the sun is the vertex of the hyperbola. If a comet follows a hyperbolic path
that is 2 trillion miles away from the sun at its closest point, and the path it is on as it approaches the
sun is perpendicular to the path it takes when it passes the sun, find an equation for the path of the
comet.