Exercises on Hyperbolae Centered at the Origin
1. For the hyperbola
x2
y2
-
= 1
9
4
the vertices are at the points ______________ and ______________ .
2. For the hyperbola
y2
x2
-
= 1
4
9
the vertices are at the points ______________ and ______________ .
3. The asymptotes of the hyperbola
x2
y2
-
= 1
9
4
have equations ______________ and ______________ .
For exercises 4 and 5, answer true or false and explain your reasoning.
4. Hyperbolae are never graphs of functions.
5. The graph of 4x2 - 25y2 = 36 is not a hyperbola since the right hand side is not equal to 1.
For exercises 6 - 12, sketch the graph of the given hyperbola.
6.
x2
y2
-
= 1
25 9
7.
x2
y2
-
= 1
4 36
8.
y2
x2
-
= 1
36 49
9.
y2
x2
-
= 1
36 25
10.
y2
x2
-
= 1
4
11.
x2
- y2 = 1
9
12. y2
x2
-
= 1
5 16
13.
x2
y2
-
= 1
7
For exercises 14 - 33, sketch of the graph of the given hyperbola and write down the equations of the asymptotes.
14. y2
9x2
-
= 1
4
15.
x2
- 16y2 = 1
4
16. 25y2 - x2 = 1
17. y2 - 16x2 = 1
18. 4x2 - 9y2 = 1
19. 100x2 - 81y2 = 1
20. 3y2 - 10x2 = 1
21. 18x2 - 11y2 = 1
22. x2 - 4y2 = 16
23. 9y2 - x2 = 36
24. y2 - 25x2 = 25
25. 25x2 - 4y2 = 100
26. x2 - 4y2 = 9
27. 16y2 - x2 = 25
28. 25y2 - 36x2 = 9
29. 36x2 - 25y2 = 49
30. 9 x2 - 12y2 = 12
31. 4y2 - 64x2 = 8
32. y2 - 10x2 = 5
33. 6x2 - 15y2 = 9
34. A hyperbola centered at the origin has vertices (-2,0) and (2,0) and passes through the point ( 4, 7
).
Find its equation.
35. A hyperbola centered at the origin has vertices (0,-5)
and (0,5) and passes through the point
( , 10).
Find its equation.
36. Go to the site, Cool Math, and give three examples of how hyperbolae appear in the "real world".
For Exercises 37 through 42, determine if the following is a line, circle, ellipse or hyperbola.
37.
x2
y2
-
= 1
100
81
38.
x2
y
-
= 1
4
36
39.
x y
-
= 1
25 49
40.
x2
y2
+
= 1
144
9
41.
x
y2
+
= 1
64
4
42.
x2
y2
+
= 1
4
4