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Name: __________________ Period: ____
Precalculus H/GT
Worksheet: Conics
Determine an equation of a circle that satisfies the given conditions.
1. Congruent to the graph x 2 y 2 18 and translated 4 units to the right.
x 4 y2 18
2
2. Endpoints of diameter, 6, 8 and 4, 2 .
x 1 y 3 50
2 2
3. Contains the points 4,5 , 2, 1 and 0,3 .
x 5 y 2 50
2 2
4. Center 1, 2 , tangent to the line 3x 2 y 6 .
x 1 y 2 1 13
2 2
Determine whether the graph of each equation represents a circle, a point, or the empty set.
5. x 2 y 2 2 x 5 y 10 0
Empty set
6. x 2 y 2 8 x 8 y 32 0
A single point
Determine an equation in standard form for each parabola described by the given geometry.
7. Focus 4,1 , directrix x 2
1
x y 1 3
2
4
8. Vertex 5, 2 , directrix y 0
1
y x 5 2
2
8
9. Vertex 2,3 , Focus 2,3 1
8
y 2 x 2 3
2
Determine the equation of an ellipse in standard form that satisfies the given conditions.
10. Focus 4,1 and 8,1 and major axis of length 10
x 6 y 1
2 2
1
25 21
11. Foci 2,5 and 2,3 and one vertex 2,9
x 2 y 4
2 2
1
24 25
-1-
Name: __________________ Period: ____
x y x2 y 2
12. Use a graphing utility, graph the equation 1 and 1 on the same set of
3 2 9 4
axes and compare the results.
[Calculator]
Challenge (No Calculator):
13. The area of the Rectangle ABCD is 2006 square units. An ellipse has area 2006 passes
through points A and C, and has foci at B and D. What is the perimeter of Rectangle ABCD?
(The area of an ellipse is given by AREAellipse ab , where a and b are the lengths of major
and minor radii)
[not to be announced]
Applications of Conics (some problems require calculator)
14. The reflector of a flashlight is in the shape of a paraboloid of revolution. Its diameter is 4
inches and its depth is 2 inch. How far from the vertex should the light bulb be placed so that
the rays will be reflected parallel to the axis?
0.5 inch from vertex
15. The University of Southern California (USC) is located about 4 kilometers west and about 4.5
kilometers south of downtown Los Angeles. A seismograph on the campus indicated that an
earthquake occurred and it is estimated that the epic center of the quake was about 60
kilometers from the university. Assume that the center of a coordinate plane is located at the
center of Los Angeles. Write an equation of the set of points that could be the epic center of
the quake.
x 4 y 4.5 3600
2 2
16. Three tracking stations have detected an earthquake in an area. The first station is located at
the origin on the map. Each grid in the map represents one square mile. The second tracking
station is located at (0, 30) while the third station is located at (35, 18). The epic center was
50 miles from the first station, 40 miles from the second station, and 13 miles from the third
station. Where was the epic center of the earthquake?
At the point (40, 30)
17. The reflecting telescope contains a mirror shaped like a paraboloid of revolution. If the
mirror is 4 inches across at its opening and is 3 feet deep, where will the collected light be
concentrated?
1
inches from the vertex
36
18. The radar for a county airport control tower is located at (5, 10) on a map. It can detect a
plane up to 20 miles away. Write an equation for the outside limits that a plane can be
detected.
x 5 y 10 400
2 2
-2-
Name: __________________ Period: ____
19. A bridge is built in the shape of a semielliptical arch. The bridge has a span of 120 feet and a
maximum height of 25 feet. Choose a suitable rectangular coordinate system and find the
height of the arch at distances 10, 30, and 50 feet from the center.
25 35 25 3 25 11
24.650 ; 21.651 ; 13.819
6 2 6
20. The control tower for the Metro Blairsville Airport is located at (9, 23) on the country map.
The radar used at the airport can detect airplanes up to 45 miles away. Write the equation for
the position of the most distant plane that the control tower can detect in terms of the country
map. Is a plane at (15, –30) inside or outside the range of the radar?
x 9 y 23 2025 , (15, –30) is outside the range.
2 2
The following problems are related to Hyperbolas and will need to be completed at a later date.
Determine the standard equation for each hyperbola, and sketch the graphs.
21. Foci 5,3 and 5,9 with transverse axis of length 4
y 6 x 5
2 2
1
4 5
22. Center 2,1 with one vertex at 3,1 and one focus at 4,1
x 2 y 1
2 2
1
25 11
23. Vertices 1,10 and 1, 2 , one focus 1,13
y 6 x 1
2 2
1
16 33
24. Two Coast Guard stations are located 600 miles apart at points A 0,0 and B 600,0 . A
distress signal from a ship is received at slightly different times by the two stations. It is
determined that the ship is 200 miles farther from station A than it is from station B. Find the
equation of a hyperbola that passes through the location of the ship.
x 300
2
y2
1
10000 80000
-3-
