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

\[2ae = 16\]

\[ \Rightarrow ae = 8\]

\[ \Rightarrow a = \frac{8}{\sqrt{2}} = 4\sqrt{2}\]

\[ \Rightarrow a^2 = 32\]

Now,

\[\left( ae \right)^2 = a^2 + b^2 \]

\[ \Rightarrow \left( 8 \right)^2 = 32 + b^2 \]

\[ \Rightarrow 64 - 32 = b^2 \]

\[ \Rightarrow b^2 = 32\]

Therefore, the equation of the hyperbola is given by

\[\frac{x^2}{32} - \frac{y^2}{32} = 1\]

\[ \Rightarrow x^2 - y^2 = 32\]