Question

Taking axes of hyperbola as coordinate axes, find its equation when the distance between the foci is $16$ and eccentricity is $\sqrt{2}$.

• A. $x^2-y^2=32$
• B. $x^2-y^2=64$
• C. $x^2-y^2=8$
• D. $x^2-y^2=16$

Answer 1

The correct option is A $x^2-y^2=32$

Given, $2ae=16$ and $e=\sqrt{2}$
$\Rightarrow a=\frac{16}{2\sqrt{2}}$
$\Rightarrow a=4\sqrt{2}$
Also, $e=\sqrt{1+\frac{b^2}{a^2}}$
$\Rightarrow e^2=1+\frac{b^2}{a^2}$
$\Rightarrow 2=1+\frac{b^2}{32}$
$\Rightarrow 1=\frac{b^2}{32}$
$\Rightarrow b^2=32\Rightarrow b=4\sqrt{2}$
$\therefore$ Required equation of hyperbola is
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\Rightarrow \frac{x^2}{32}-\frac{y^2}{32}=1$
$\Rightarrow x^2-y^2=32$.