Review Sheet #8
Determine the domain and range of the conic section describe by the following
equations.
1. 4 x2  16 y 2  4 x  96 y  11  0
2. 18x2  18 y 2  12 x  72 y  250  0
3. 2 y 2  x  12 y  16  0
4. 144 x2  900 y 2  96 x  900 y  3391  0
Use the discriminant and identify the conic Rewrite each equation as several
functions of y. Graph them simultaneously on your graphing calculator. Sketch the
graph from the viewing window.
5. 2 x 2  xy  3 y 2  3x  4 y  6  0
6.  x 2  3xy  4 y 2  5 x  10 y  20  0
7. 2 x 2  4 xy  8 y 2  10 x  4 y  13  0
8. 2 x 2  4 xy  2 y 2  5x  6 y  15  0
Write the equation of the conic described by each of the following.
9. Traced by a point that moves such that the sum of its distances to  3,8 and  5,8 is
always 14.
10. With the property that the distance between the point  1, 2  and any point P  x, y 
on the curve is equal to the distance between P and the line y  6 .
Solve the following application problems. Make sure to include a sketch with your
work.
11. One of the world’s largest reflecting telescopes is called the Hale telescope in honor
of the American astronomer George E. Hale. It is located at the Palomar Mountain
Observatory northeast of San Diego, California. It uses a mirror 200 inches in
diameter. A cross section of the mirror through a diameter is a parabola. The
distance from the focus to the vertex is 666 inches.
a. Sketch the parabola.
b. Find the equation of the parabola
c. What is the depth of the parabolic cross section of the mirror?
12. A door in the shape of a parabolic arch is 12 feet high at the center and 5 feet wide at
the base. A rectangular box 9 feet tall is to be slid through the door. What is the
widest the box can be?
13. A semi-elliptical arch over a tunnel for a road through a mountain has a major axis of
80 feet and a height at the center of 30 feet. Find an equation of the elliptical tunnel.
Determine the height of the arch 5 feet from the edge of the tunnel.
14. An arch in the form of a semi-ellipse is 40 feet wide and 15 feet high at the center.
Find the height of the arch at 10 feet from the center.
15. Two LORAN stations are positioned 100 miles apart along a straight shore. Their
coordinates are  50, 0  and  50,0 . A ship records a time difference of 0.00032
seconds between the LORAN signals. If the ship is 20 miles offshore (that is,
y  20 ) when the signals are recorded, find one of the possible coordinates of the
ship? (Note: the speed of each radio signal is 186,000 miles per second.)
16. You’ve lost the assembly directions for your 12 ft. diameter satellite antenna, but you
know you’re okay because it’s parabolic, and you get parabolas. Lying flat on the
ground, it measures 2 ft. high. Where should you locate the satellite dish’s receiver
(i.e. how far above the vertex is it)? Is it inside or outside the “dish”?
17. You are constructing a parabolic arch over a 680 ft. wide, 200 ft. deep river gorge
upon which you will support a bridge. You know you will have the most support if
the focal point is on the river surface. Will the arch be even with the top of the
gorge? Show the analysis that leads to your conclusion.
18. You shoot an army-surplus mortar shell at your satellite dish, because you couldn’t
get it to work, and blowing it up seems more fun than taking it apart. The mortar
shell follows a parabolic trajectory from 500 ft. away from the dish. It achieves an
altitude of 100 ft. at the highest point and scores a direct hit on the receiver antenna.
Is the focal point of the parabolic trajectory above ground or below ground, and how
far?
19. The receiver in a parabolic television dish antenna is 3 feet from the vertex and is
located at the focus (see figure). Find an equation of a cross section of the reflector.
(Assume that the dish is directed upward and the vertex is at the origin.)
20. A solar collector for heating water is constructed with a sheet of stainless steel that is
formed into the shape of a parabola (see figure). The water flows through a pipe that
passes through the focus of the parabola. At what distance is the pipe from the
vertex?
21. A commemorative parabolic steel arch is planned, with its axis vertical and its feet
240 meters apart. If the focus of the parabola is to be 64 meters above the ground,
how high must the arch be?
22. The orbit of a satellite about the Earth is elliptical with the center of the earth acting
as one of the foci of this ellipse. Assume that the earth is a perfect sphere with a
diameter of 8000 miles. The minimum altitude (perigee) of the satellite is 100 miles
and the maximum altitude (apogee) is 1200 miles. Write the equation of this orbit
assuming that the center of the earth is the origin of the coordinate system. (Hint: the
apogee and perigee both occur when the satellite is a t an endpoint of the major axis.)
23. The Earth moves in an elliptical orbit about the Sun with the Sun at one focus. If the
least distance Earth to Sun is 91,340,000 miles and the greatest is 94,450,000 miles,
how far is the Sun from the other focus.
24. Each cable of suspension bridge is suspended (in the shape of a parabola) between
two towers that are 400 feet apart and 50 feet above the roadway (see figure). The
cables touch the roadway midway between the towers.
a. Find an equation for the parabolic shape of each cable.
b. Find the length of the vertical supporting cable when
x  100 .
Review Sheet #8 Answers
1.
3.
1
 1

D :    39 ,  + 39 
2
 2


39
39 
R :  3 
, 3

2
2 

D : (, )
D : [2,  )
4.
R : (,  )
5. Ellipse
9.
1
1

D :   3 2,
 3 2
3
3

2.
R :  2  3 2,  2  3 2 
 x  1
6. Hyperbola
2

 y  8
2
1
R : (,
3
5
] [ , )
2
2
7. Ellipse
8. Parabola
10.  x  1  8  y  4 
2
49
33
11.
(a) sketch
(b) y 2  2664 x
(if parabola opens right)
2
(if parabola opens left)
y  2664 x
(if parabola opens up)
x 2  2664 y
2
(if parabola opens down)
x  2664 y
(c) depth = 3.75 inches
12. width = 2.5 feet
13. height = 14.5 feet
14. height = 12.99 feet
15.  33.24, 20  or  33.24, 20 
16. 4.5 feet up from vertex; “outside” of the dish
17. No, the arch will be below the top of the gorge. Construct parabolic arch 170 feet
high with 30 feet of support beams to reach the necessary 200 feet.
18. Below ground by 56¼ feet.
19. x 2  12 y
20. distance = 2¼ feet
21. The arch should be 100 m above the ground.
(x  4550)2
y2

1
22.
21622500 92000
23. 3,110,000 miles
24. x 2  800y, 12.5 feet