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Equations for Some Hyperbolas 831
Lesson
Lesson 12-
BIG IDEA From the geometric definition of a hyperbola, an equation for any hyperbola symmetric to the x- and y-axes can be found.
The edges of the silhouettes of each of the towers pictured at the right are parts of hyperbolas. Structures with this shape are able to withstand higher winds and require less material to build than any other form.
What Is a Hyperbola?
Like an ellipse, a hyperbola is determined by two foci and a focal constant. However, instead of a constant sum of distances from the foci, a point on a hyperbola must be at a constant difference of distances from the foci. The following Activity shows one way to find points on a hyperbola.
MATERIALS conic graph paper with 6 units between the centers of the circles Step 1 Copy the foci and points P 1 and P 2 at the right. Find P 1 F 1 , P 2 F 1 , P 1 F 2 , and P 2 F 2 , then calculate P 1 F 1 - P 1 F 2 and P 2 F 1 - P 2 F 2. Do both differences equal the same constant? Step 2 Plot two more points P (^) n such that P (^) n F 1 = 8 and P (^) n F 2 = 6, and then two more such that P (^) n F 1 = 7 and P (^) n F 2 = 5. Continue this process to find four more points such that P (^) n F 1 - P (^) n F 2 is always 2. (continued on next page)
ActivityActivity
F 1 F 2
P 2
(^6) P 1 7 8
1 1
2
2
3
3
4
4
5
65 8 7
F 1 F 2
P 2
(^6) P 1 7 8
1 1
2
2
3
3
4
4
5
65 8 7
Vocabulary hyperbola foci, focal constant of a hyperbola vertices of a hyperbola standard position of a hyperbola standard form of an equation for a hyperbola
Mental Math
Suppose a function f contains the points (4, 17), (9, 12), and (13, 13). a. Find the rate of change from (4, 17) to (9, 12). b. Find the rate of change from (9, 12) to (13, 13). c. Could the graph of f be a line?
Equations for
Some Hyperbolas
832 Quadratic Relations
Chapter 12
Step 3 Repeat Step 2, plotting ten points Pn such that Pn F 2 - Pn F 1 = 2. Step 4 Draw a smooth curve through the points you plotted in Step 2, and another through the points you plotted in Step 3. These are two branches of a hyperbola. The branches do not intersect.
In general, if d is a positive number less than F 1 F 2 , the set of all points P such that | PF 1 - PF 2 | = d is a hyperbola. The absolute value means that the hyperbola has two branches, one from PF 1 - PF 2 = d , and the other from PF 1 - PF 2 = – d. The absolute value function allows both branches to be described with one equation.
Definition of Hyperbola
Let F 1 and F 2 be any two points and d be a constant with 0 < d < F 1 F 2. Then the hyperbola with foci F 1 and F 2 and focal constant d is the set of points P in a plane that satisfy |PF 1 - PF 2 | = d.
The vertices V 1 and V 2 of the hyperbola are the
intersection points of F 1 F^ 2 and the hyperbola.
While it may look like each branch of the hyperbola is a parabola, this is not the case. Each branch of a hyperbola has asymptotes. In the figure at the right, 1 and 2 are asymptotes. The farther points on the hyperbola are from a vertex of the hyperbola, the closer they are to an asymptote, without ever touching. In contrast, parabolas do not have asymptotes.
The Standard Form of an Equation for
a Hyperbola
A hyperbola is in standard position if it is centered at the origin with its foci on an axis. An equation for a hyperbola in standard position resembles the standard form of an equation for an ellipse.
Equation for a Hyperbola Theorem
The hyperbola with foci (c, 0) and (– c, 0) and focal constant 2a has equation x _^2 a^2 -^
_y^2 b^2 =^ 1, where^ b
(^2) = c (^2) - a (^2).
F 1 F 2
1 2
V 1
P
V 2
vertex vertex
| PF 1 -^ PF 2 | =^ d
branch branch
asymptote asymptote
F 1 F 2
1 2
V 1
P
V 2
vertex vertex
| PF 1 -^ PF 2 | =^ d
branch branch
asymptote asymptote
834 Quadratic Relations
Chapter 12
Then x^2 - y^2 = 1. The hyperbola with this equation is symmetric to both axes. Consequently, each point on the hyperbola in the first quadrant has reflection images on the hyperbola in other quadrants. The graph at the right shows the reflection images of A , B , C , and D over the x -axis and the y -axis.
A = (1, 0)
B = (2, √3 )^ ≈ (2, 1.73)
C = (3, √8 )^ ≈ (3, 2.83)
D = (4, √15 )^ ≈ (4, 3.87)
The lines y = – x and y = x appear to be the asymptotes of x^2 - y^2 = 1. We can verify the equations for the asymptotes algebraically.
When x^2 - y^2 = 1,
y^2 = x^2 - 1.
So y = ± √ x^2 - 1.
As values of x get larger, √ x^2 - 1 becomes closer to √ x^2 , which
is | x |. However, because √ x^2 - 1 ≠ √ x^2 , the curve x^2 - y^2 = 1 never
intersects the lines with equations y = x or y = – x. So, y gets closer to x or – x but never reaches it.
According to the Graph Scale-Change Theorem, the scale change
S (^) a, b maps x^2 - y^2 = 1 onto x _^2 a^2
- y _^2 b^2 = 1. Under the same scale
change, the asymptotes y = ± x of x^2 - y^2 = 1 are mapped onto the lines with equations y _ b = ± _ ax. These lines are the asymptotes of _^ x^2 a^2
Asymptotes of a Hyperbola Theorem
The asymptotes of the hyperbola with equation x _^2 a^2
- y^ _^2 b^2 = 1 are _^ y b = ±^
_x a , or^ y^ =^ ±^
b_ a x.
QY
x
y
(4, √ 15 )^ (4, √ 15 ) =^ D
( 4, √ 15 ) (4, √ 15 )
(3, √⎯ 8 ) (3, √⎯ 8 ) = C (2, √⎯ 3 ) (2, √⎯ 3 ) = B
(2, √⎯ 3 ) (3, √⎯ 8 )
( 2, √⎯ 3 ) ( 3, √⎯ 8 )
(1, 0) (1, 0) =^ A
asymptote y =^ x asymptote y =^ x
x^2 - y^2 = 1
x
y
(4, √ 15 )^ (4, √ 15 ) =^ D
( 4, √ 15 ) (4, √ 15 )
(3, √⎯ 8 ) (3, √⎯ 8 ) = C (2, √⎯ 3 ) (2, √⎯ 3 ) = B
(2, √⎯ 3 ) (3, √⎯ 8 )
( 2, √⎯ 3 ) ( 3, √⎯ 8 )
(1, 0) (1, 0) =^ A
asymptote y =^ x asymptote y =^ x
x^2 - y^2 = 1
QY What are the asymptotes of the hyperbola in Example 1?
QY What are the asymptotes of the hyperbola in Example 1?
Equations for Some Hyperbolas 835
Lesson 12-
Graphing a Hyperbola with Equation in
Standard Form
To graph x _^2 a^2
- y _^2 b^2 = 1 by hand, notice that ( a , 0) and
(– a , 0) satisfy the equation. These are the vertices of the hyperbola. When x = 0, y is not a real number, so the hyperbola does not intersect the y -axis. Use the asymptotes to make an accurate sketch of the graph. Remember that the asymptotes are not part of the hyperbola.
Example 2
Graph the hyperbola with equation x __^2 16 -^
__y^2 36 =^ 1. Solution The equation is in standard form. So, a 2 = 16 and a = 4. The vertices are (4, 0) and ( – 4, 0). The asymptotes are _y 6 =^ ±^
_x 4 , or^ y^ =^ ±^
3 __
2 x. Carefully graph the vertices and asymptotes. Then sketch the hyperbola.
Check Solve x _^2 16 -^
_y^2 36 =^ 1 for^ y^ on a CAS. One CAS solution is shown below.
The complete solution is
y = 3 ·^ _____√x^2 -^16 2 and^ x
(^2) - 16 ≥ 0 or y = – _____^3 ·^ √x^2 -^16 2 and^ x^
So y =^3
_________√x^2 -^^16
2 or y^ =^ –^
_________^3 √x^2 -^^16
Graph both equations on the same axes on a graphing utility. Although the graphing utility may have trouble graphing values close to the vertices of the hyperbola, the output closely resembles the hand-drawn solution.
x
y
( a , 0) ( c , 0) ( c , 0)
( a , 0)
y = bax y = bax
x
y
( a , 0) ( c , 0) ( c , 0)
( a , 0)
y = bax y = bax
y
x 8 6 4 2 2 4 6 8
4 2
6
8
4
2
6 8
(4, 0) (4, 0)
y =^32 x y =^ 32 x
y
x 8 6 4 2 2 4 6 8
4 2
6
8
4
2
6 8
(4, 0) (4, 0)
y =^32 x y =^ 32 x
Equations for Some Hyperbolas 837
Lesson 12-
REVIEW
- In Australia, a type of football is played on elliptical fields. One such field has a major axis of length 185 meters and minor axis of length 155 meters. Surrounding it is an elliptical fence with major axis of length 187 meters and minor axis of length 157 meters. The 1-meter wide track between the fence and the field is to be covered with turf. Find the area of the track. (Lesson 12-5)
In 15 and 16, graph the ellipse with the given equation. (Lesson 12-4)
- x _^2 4 +^
_ y^2 25 =^1 16.^
x _^2 9 +^ y
- Standard Quonset huts are semicircular with a diameter of 20 feet and a length of 48 feet. (Lesson 12-2) a. Inside the hut, how close to either side of the hut could a 6-foot soldier stand upright? b. What is the volume of a hut?
- An auto dealer is having a Fourth of July extravaganza. The dealership plans to be open for 72 hours straight. Suppose the dealer has 100 new cars on the lot and is able to sell an average of 4 cars every 3 hours. (Lesson 3-1) a. Let h be the number of hours the car dealership has been open and let C be the number of cars remaining on the lot. Find three other pairs of values that satisfy this relation and complete the table. b. Write a formula for the number of cars C on the lot as a function of the number of hours h the sale has been on. c. After how many hours will there be only 60 cars left? d. If the dealership is able to maintain the pace of 4 cars sold every 3 hours, will the dealer sell all the cars on the lot during the sale? How can you tell?
EXPLORATION
- The words ellipsis and hyperbole have literary meanings. What are these meanings?
- In Round the Moon , a novel written by Jules Verne in 1870, a group of men launch a rocket to the Moon. During the journey they argue whether the rocket trajectory is hyperbolic or parabolic. Because each curve is infinite, the men believe they are doomed to travel infinitely through space. Find out on which trajectory modern day rockets travel and whether or not the men had reason to worry.
1 m
1 m
1 m Field 1 m
Fence not to scale
1 m
1 m
1 m Field 1 m
Fence not to scale
During World War II, easy-to- build Quonset huts were used as barracks for troops.
During World War II, easy-to- build Quonset huts were used as barracks for troops.
h^0 3?^?^? C 100 96???
h^0 3?^?^? C 100 96???
QY ANSWER y = ± 3 _ 4 x
