Question

The distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2}.$
Then the equation of the hyperbola is $x^2-y^2={2k}^2.$ Find k.

Answer 1

$2ae=16\therefore ae=8,e=\sqrt{2}$ $\therefore a=4\sqrt{2}$ or $a^2=32$
$b^2=a^2(e^2-1)=32(2-1)=32.$
Hence the hyperbola is $x^2-y^2=32.$
Ans: 4