Hyperbolas in Standard Form

Show your work for each problem.

1.
Which equation represents a hyperbola with a horizontal transverse axis?
1. a)
2. b)
3. c)
4. d)
Answer:
is correct. When has the positive coefficient, the hyperbola opens left and right (horizontal transverse axis).
2.
For the hyperbola , what is the value of ?
1. a)
2. b)
3. c)
4. d)
Answer:
Since is positive, , so . Remember: is always under the positive term.
3.
What is the center of the hyperbola ? Enter as
Answer: (3, -2)
From , we get . From , we get . So the center is .
4.
For a hyperbola with and , what is ?
Answer: 5
, so .
5.
What are the vertices of the hyperbola ?
1. a) and
2. b) and
3. c) and
4. d) and
Answer: and
This is a horizontal hyperbola with center and . Vertices are at , which are and .
6.
What is the slope of the asymptotes for ? Enter as a fraction.
Answer: 3/4
With and , we have and . For a horizontal hyperbola, slope .
7.
What are the foci of ?
1. a) and
2. b) and
3. c) and
4. d) and
Answer: and
, so . This is a horizontal hyperbola, so foci are at .
8.
Find the center, vertices, and foci of
Answer: Center: (-1, 2), Vertices: (-3, 2) and (1, 2), Foci: (-4, 2) and (2, 2)
1. What is the center ? (-1, 2)
2. What is ? 2
3. What is ? sqrt(5)
4. What is ? 3
9.
Write the standard form equation for a hyperbola with center , vertex at , and focus at . Enter as y^2/a - x^2/b = 1
Answer: y^2/16 - x^2/9 = 1
Vertical hyperbola with , . . Equation: .
10.
What is the equation of the asymptotes for ?
1. a)
2. b)
3. c)
4. d)
Answer:
Center , , . Vertical hyperbola, so slope . Asymptotes: .
11.
Convert to standard form.
Answer: (x-2)^2/4 - (y+1)^2/9 = 1
1. Group x and y terms: 4
2. Complete square for x: 4
3. Complete square for y: 1
4. After completing squares, right side = 36
12.
A hyperbola has foci at and , and the difference of distances from any point on the hyperbola to the foci is 8. What is ?
Answer: 4
For a hyperbola, . Given the difference is 8, we have , so .
13.
Write the equation of the hyperbola with foci at and passing through the point .
Answer: y^2/10 - x^2/15 = 1
1. What is ? 5
2. Distance from to is sqrt(10)
3. Distance from to is 3sqrt(10)
4. What is ? 2sqrt(10)
5. So 10
6. What is ? 15
14.
The eccentricity of a hyperbola with and is:
1. a)
2. b)
3. c)
4. d)
Answer:
, so . With : . Note: For hyperbolas, always.
15.
Find all key features of
Answer: Center (1,-2), a=6, b=8, c=10, Vertices (-5,-2) and (7,-2), Foci (-9,-2) and (11,-2), Asymptotes y+2=+/-4/3(x-1)
1. What is the center? (1, -2)
2. What is ? 6
3. What is ? 8
4. What is ? 10
5. What is the asymptote slope? 4/3