Suggested languages for you:

Americas

Europe

Q. 85

Expert-verified
Found in: Page 669

### Precalculus Enhanced with Graphing Utilities

Book edition 6th
Author(s) Sullivan
Pages 1200 pages
ISBN 9780321795465

# A hyperbola for which $a=b$ is called an equilateral hyperbola. Find the eccentricity $e$ of an equilateral hyperbola. [Note: The eccentricity of a hyperbola is defined in Problem 81.]

The eccentricity of a equilateral hyperbola is $\sqrt{2}$

See the step by step solution

## Step 1. Eccentricity

Eccentricity is a number that is calculate in the following way

$e=\frac{c}{a}$

where $a$ is the distance from the center to the vertex and $c$ is the distance from the center to the focus.

## Finding the eccentricity

For an equilateral hyperbola, it hold that $a=b$

For any hyperbola the following formula holds:

${b}^{2}={c}^{2}-{a}^{2}$

therefore

${c}^{2}={a}^{2}+{b}^{2}$

Since, $a=b$, you can write previous equation as

${c}^{2}={a}^{2}+{a}^{2}=2{a}^{2}$

then

$c=a\sqrt{2}$

## Finding the eccentricity

Put $c=a\sqrt{2}$ in the eccentricity formula to get

$e=\frac{a\sqrt{2}}{a}\phantom{\rule{0ex}{0ex}}e=\sqrt{2}$

## Step 4. Conclusion

The eccentricity of a equilateral hyperbola is $\sqrt{2}$