Question

Find the equation to the hyperbola, referred to its axes as axes of coordinates, the distance between whose foci is $16$ and whose eccentricity is $\sqrt{2}$.

Answer 1

Standard equation

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

x axis is the transverse axis of the parabola

Given :

Distance between foci i.e,

$2ae=16$

$ae=8$

Also, $e=\sqrt{2}$

$\therefore a=\frac{8}{\sqrt{2}}$

$a=4\sqrt{2}$

and

$e^2=1+\frac{b^2}{a^2}$

$2=1+\frac{b^2}{32}$

$b^2=32$

Thus, the equation of hyperbola is

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