Question

Write the equation of the hyperbola of eccentricity $\sqrt{2}$.if it is know that the distance between its foci is 6.

Answer 1

$Eccentricitye=\sqrt{2}$

$Distancebetweenfociis$
$2ae=16$

$2a\sqrt{2}=16$

$a=\frac{16}{2\sqrt{2}}=4\sqrt{2}$

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

$\sqrt{2}=\frac{\sqrt{32+b^2}}{4\sqrt{2}}$

$8=\sqrt{32+b^2}$

$64=32+b^2$ $b^2=32$

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

$Rewritingweget,x^2-y^2=32$