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.