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

Answer 1

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

The distance between the foci,
$2ae=16\Rightarrow a=\frac{16}{2\sqrt{2}}=4\sqrt{2}$
Eccentricity,
$e^2=1+\frac{b^2}{a^2}\Rightarrow 2=1+\frac{b^2}{32}\Rightarrow b=4\sqrt{2}$
Hence, the equation of hyperbola
$\frac{x^2}{(4\sqrt{2}{)}^2}-\frac{y^2}{(4\sqrt{2}{)}^2}=1$
$\Rightarrow x^2-y^2=32$