Question

If the distance between the foci of a hyperbola is 16 and its ecentricity is $\sqrt{2}$, then obtain its equation.

Answer 1

We have
$$\begin{gathered} 2ae=16 \\ \Rightarrow ae=8 \\ \Rightarrow a=\frac{8}{\sqrt{2}}=4\sqrt{2} \\ \Rightarrow a^2=32 \end{gathered}$$
Now,
$$\begin{gathered} {\left(ae\right)}^2=a^2+b^2 \\ \Rightarrow {\left(8\right)}^2=32+b^2 \\ \Rightarrow 64-32=b^2 \\ \Rightarrow b^2=32 \end{gathered}$$

Therefore, the equation of the hyperbola is given by
$$\begin{gathered} \frac{x^2}{{\displaystyle 32}}-\frac{y^2}{{\displaystyle 32}}=1 \\ \Rightarrow x^2-y^2=32 \end{gathered}$$