Question

Write the equation of the hyperbola of eccentricity $\sqrt{2}$, if it is known that the distance between its foci is 16.

Answer 1

The foci of the hyperbola are of the form $\left(ae,0\right)$ and $\left(-ae,0\right)$.

$$\begin{gathered} \mathrm{Distance}\mathrm{between}\mathrm{the}foci=\sqrt{{\left(ae-(-ae\right)}^2+0^{{}^2}} \\ =\sqrt{{\left(2ae\right)}^{{}^2}} \\ =2ae \end{gathered}$$
Distance between the foci is 16 and eccentricity of the hyperbola is $\sqrt{2}$.

$$\begin{gathered} \therefore 2ae=16 \\ \Rightarrow 2\sqrt{2}a=16 \\ \Rightarrow a=4\sqrt{2} \end{gathered}$$

$$\begin{gathered} \mathrm{Now,}{b}^2=a^2(e^2-1) \\ \Rightarrow b^2={\left(4\sqrt{2}\right)}^2\left(\right(\sqrt{2}{)}^2-1) \\ \Rightarrow b^2=32 \end{gathered}$$

Equation of the hyperbola is given below:
$$\begin{gathered} \frac{x^2}{{\left(4\sqrt{2}\right)}^2}-\frac{y^2}{32}=1 \\ \Rightarrow \frac{x^2}{32}-\frac{y^2}{32}=1 \end{gathered}$$