Question

Find the equation of the hyperbola, referred to its principal axes as axes of coordinates, in following cases :
The distance between the foci $=16$ and eccentricity $=\sqrt{2}$

Answer 1

Given that, $e=\sqrt{2}$

$\because$ Distance between the foci $=16$

$\Rightarrow 2ae=16$

$\Rightarrow 2a\times \sqrt{2}=16$

$\Rightarrow a=4\sqrt{2}$

Given that $e=\sqrt{2}$ So, hyperbola is a rectangular hyperbola.

Equation of rectangular hyperbola

$x^2-y^2=a^2$

$\Rightarrow x^2-y^2=(4\sqrt{2}{)}^2$

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

Therefore, equation of hyperbola is $x^2-y^2=32$