Introduction to Hyperbolas

Show your work for each problem.

1.
Which equation represents a hyperbola?
1. a)
2. b)
3. c)
4. d)
Answer:
is a hyperbola because it has subtraction between the squared terms. The first option is an ellipse (addition), the third is a circle, and the fourth is a parabola.
2.
The hyperbola opens in which direction?
1. a)Up and down
2. b)Diagonally
3. c)Left and right
4. d)It does not open
Answer: Left and right
Since is the positive term (comes first and has no negative sign), this is a horizontal hyperbola that opens left and right.
3.
For the hyperbola , what is the value of ?
Answer: 5
In the standard form, is always under the positive term. Here , so .
4.
For the hyperbola , what is the value of ?
Answer: 4
In the standard form, is under the negative term. Here , so .
5.
What is the correct formula to find (focal distance) in a hyperbola?
1. a)
2. b)
3. c)
4. d)
Answer:
For hyperbolas, use . This is different from ellipses where . In hyperbolas, because the foci are farther from the center than the vertices.
6.
For the hyperbola , calculate the value of .
Answer: 5
, so .
7.
Find the vertices of the hyperbola .
Answer: (-6, 0) and (6, 0)
1. Is this a horizontal or vertical hyperbola? horizontal
2. What is ? 36
3. What is ? 6
4. For a horizontal hyperbola, vertices are at . What are the vertices? (-6, 0) and (6, 0)
8.
Find the foci of the hyperbola .
Answer: (0, -5) and (0, 5)
1. Is this a horizontal or vertical hyperbola? vertical
2. What are and ? a^2 = 9, b^2 = 16
3. Calculate . 25
4. What is ? 5
5. For a vertical hyperbola, foci are at . What are the foci? (0, -5) and (0, 5)
9.
What are the asymptotes of the hyperbola ?
1. a)
2. b)
3. c)
4. d)
Answer:
This is a horizontal hyperbola with () and (). For horizontal hyperbolas, asymptotes are .
10.
Find the asymptotes of the hyperbola .
Answer: y = (5/4)x and y = -(5/4)x
1. Is this a horizontal or vertical hyperbola? vertical
2. What is ? 5
3. What is ? 4
4. For vertical hyperbolas, asymptotes are . What is the slope? 5/4
11.
Find the center, vertices, foci, and asymptotes of .
Answer: Center (0,0), Vertices (+-7,0), Foci (+-sqrt(73),0), Asymptotes y = +-sqrt(24)/7 x
1. What is the center? (0, 0)
2. What is ? 7
3. What are the vertices? (-7, 0) and (7, 0)
4. Calculate . 73
5. What are the foci? (+-sqrt(73), 0)
6. What is ? sqrt(24)
7. What are the asymptotes? y = +-sqrt(24)/7 x
12.
A hyperbola has vertices at and , and foci at and . What is the value of ?
Answer: 8
Vertices at means . Foci at means . Using : , so and .
13.
The defining property of a hyperbola is that for any point on the curve:
1. a)The sum of distances to the foci is constant
2. b)The difference of distances to the foci is constant
3. c)The product of distances to the foci is constant
4. d)The ratio of distances to the foci is constant
Answer: The difference of distances to the foci is constant
A hyperbola is defined by the constant absolute difference of distances: . An ellipse has constant sum. This is the key distinction.
14.
Write the equation of a hyperbola with vertices at and foci at .
Answer: x^2/25 - y^2/144 = 1
1. Is this a horizontal or vertical hyperbola? horizontal
2. What is ? 5
3. What is ? 13
4. Use to find . 144
5. Write the equation. x^2/25 - y^2/144 = 1
15.
GPS systems determine location using the intersection of hyperbolas. Why are hyperbolas used instead of circles?
1. a)Circles are too difficult to calculate
2. b)GPS measures the difference in arrival times between satellite signals
3. c)GPS measures exact distances from satellites
4. d)Hyperbolas are more accurate than circles
Answer: GPS measures the difference in arrival times between satellite signals
GPS receivers measure the time difference between signals from two satellites. This time difference corresponds to a difference in distances, which defines a hyperbola. The intersection of multiple hyperbolas pinpoints your location.