Question

The distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2}$ , then find the equation of hyperbola.

Answer 1

As $e=\sqrt{2}$ and $2ae=16$

Thus,$\sqrt{2}a=8$

$\Rightarrow a=\frac{8}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}=\frac{8\sqrt{2}}{2}=4\sqrt{2}$

Thus,$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$\Rightarrow \frac{x^2}{{\left(4\sqrt{2}\right)}^2}-\frac{y^2}{{\left(4\sqrt{2}\right)}^2}=1$

$\Rightarrow \frac{x^2}{32}-\frac{y^2}{32}=1$

or $x^2-y^2=32$