Question

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.]

Short answer

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

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

$c^2=a^2+a^2=2a^2$

then

$c=a\sqrt{2}$

Finding the eccentricity

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

$$\begin{gathered} e=\frac{a\sqrt{2}}{a} \\ e=\sqrt{2} \end{gathered}$$

Step 4. Conclusion

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