Question

Write the equation of the hyperbola whose vertices are (± 3, 0) and foci at (± 5, 0).

Answer 1

The vertices of the hyperbola and the foci are \[\left( \pm a, 0 \right)\] and \[\left( \pm ae, 0 \right)\], respectively. 

\[\therefore a = 3 \]

and ae = 5

Using the relation \[b^2 = a^2 \left( e^2 - 1 \right)\], we get:

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

\[ \Rightarrow b^2 = 25 - 9\]

\[ \Rightarrow b^2 = 16\]

\[ \Rightarrow b = 4\]

Therefore, the equation of hyperbola is 

\[\frac{x^2}{9} - \frac{y^2}{16} = 1\]