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.