Question

The distance between the foci of a hyperbola is $16$ and its eccentricity is $\sqrt{2}$ . Its equation is

• A. $x^2-y^2=32$
• B. $2x-3y^2=7$
• C. $\frac{x^2}{4}-\frac{y^2}{9}=1$
• D. None of these

Answer 1

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

Let the equation of hyperbola be:
$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\dots \left(1\right)$
We have eccentricity, $e=\sqrt{2}$ and distance between foci $=2ae=16$
$\Rightarrow ae=8\dots \left(2\right)$
Put the value of e in eq. $\left(2\right)$
$\Rightarrow a\times \sqrt{2}=8$
$\Rightarrow a=4\sqrt{2}$
Squaring on both sides
$\Rightarrow a^2=32$
$\because b^2=a^2(e^2-1)$
$\Rightarrow b^2=32(2-1)[\because a^2=32\text{and}e=\sqrt{2}]$
$\Rightarrow b^2=32$
Put the value of $a^2=32$ and $b^2=32$ in equation $\left(1\right)$
Hence, hyperbola is
$\frac{x^2}{32}-\frac{y^2}{32}=1$
$\Rightarrow x^2-y^2=32$

Hence, option $\left(A\right)$ is correct.