# Example of Finding Equation of Hyperbola

Example: Find an equation of the hyperbola with centre at origin, focus at $\left( {7,0} \right)$ and vertex at the point $\left( {3,0} \right)$.
Since the focus of an ellipse lies on X-axis with centre $\left( {0,0} \right)$, so let the required equation of hyperbola be
$\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1\,\,\,{\text{ – – – }}\left( {\text{i}} \right)$

For hyperbola we have the relation
${a^2} + {b^2} = {a^2}{e^2}$

By the given conditions, we have $ae = 7$, $a = 3$. Putting these values in the given condition of hyperbola, we get
$\begin{gathered} \Rightarrow {b^2} = 49 – 9 \\ \Rightarrow b{\,^2}\, = 40 \\ \end{gathered}$

Putting the values of ${a^2}$ and ${b^2}$ in the equation (i), we have
$\frac{{{x^2}}}{9} – \frac{{{y^2}}}{{40}} = 1$
This is the required equation of hyperbola.