EXAMPLE 1 Graph an equation of a hyperbola

Graph 25y2 – 4x2 = 100. Identify the vertices, foci, and asymptotes of the hyperbola.

SOLUTION

STEP 1Rewrite the equation in standard form.25y2 – 4x2 = 100 Write original equation.

25y2

100 – 4x2

100100 100= Divide each side by 100.

y2

4 –y2

25 = 1 Simplify.

EXAMPLE 1 Graph an equation of a hyperbola

STEP 2Identify the vertices, foci, and asymptotes. Note that a2 = 4 and b2 = 25, so a = 2 and b = 5. The y2 - term is positive, so the transverse axis is vertical and the vertices are at (0, +2). Find the foci.

c2 = a2 – b2 = 22 – 52 = 29.so c = 29.

The foci are at ( 0, + )29. (0, + 5.4).

The asymptotes are y = ab

+ x or 25

+ xy =

EXAMPLE 1 Graph an equation of a hyperbola

STEP 3

Draw the hyperbola. First draw a rectangle centered at the origin that is 2a = 4 units high and 2b = 10 units wide. The asymptotes pass through opposite corners of the rectangle. Then, draw the hyperbola passing through the vertices and approaching the asymptotes.

EXAMPLE 2 Write an equation of a hyperbola

Write an equation of the hyperbola with foci at (–4, 0) and (4, 0) and vertices at (–3, 0) and (3, 0).SOLUTIONThe foci and vertices lie on the x-axis equidistant from the origin, so the transverse axis is horizontal and the center is the origin. The foci are each 4 units from the center, so c = 4. The vertices are each 3 units from the center, so a = 3.

EXAMPLE 2 Write an equation of a hyperbola

Because c2 = a2 + b2, you have b2 = c2 – a2. Find b2.

b2 = c2 – a2 = 42 – 32 = 7

Because the transverse axis is horizontal, the standard form of the equation is as follows:

x2

32 –y2

7 = 1 Substitute 3 for a and 7 for b2.

x2

9 –y2

7 = 1 Simplify

GUIDED PRACTICE for Examples 1 and 2

Graph the equation. Identify the vertices, foci, and asymptotes of the hyperbola.

1. x2

16 –y2

49 = 1

SOLUTION

(+4, 0) , ( + ) ,65 , 0 74

+ xy =

GUIDED PRACTICE for Examples 1 and 2

2. y2

36 – x2 = 1

SOLUTION

(0, +6) , ( 0, + ) ,37 y = +6x

GUIDED PRACTICE for Examples 1 and 2

3. 4y2 – 9x2 = 36

SOLUTION

( 0, + ) ,13 (0, +3) , 32

+ xy =

GUIDED PRACTICE for Examples 1 and 2

Write an equation of the hyperbola with the given foci and vertices.4. Foci: (–3, 0), (3, 0)

Vertices: (–1, 0), (1, 0)

SOLUTION x2 –y2

8 = 1

5. Foci: (0, – 10), (0, 10)Vertices: (0, – 6), (0, 6)

SOLUTION = 1 y2

36 –x2

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