Question

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

Answer 1

Focii of hyperbola are $(\pm ae,0)$

Distance between them$=2ae$

$2ae=16$

$ae=8$

Given eccentricity $e=\sqrt{2}$

$a\times \sqrt{2}=8$

$a=4\sqrt{2}$

For a hyperbola, we know that

$b^2=a^2(e^2-1)$

$b^2=32(2-1)$

$b^2=32$

$a^2=32$

$\therefore$Equation of hyperbola is

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

$\therefore x^2-y^2=32$