Question

Find the equation of the ellipse in the case:

 Vertices (± 5, 0), foci (± 4, 0)

Answer 1

\[ \text{ Vertices} \left( \pm 5, 0 \right) \text{ and focus} \left( \pm 4, 0 \right)\]
\[\text{ The coordinates of its vertices and foci are } \left( \pm a, 0 \right)\text{and } \left( \pm ae, 0 \right), \text{ respectively } .\]
\[i . e . a = 5 \text{ and ae } = 4\]
\[ \therefore e = \frac{4}{5}\]
\[\text{ Now, }  b^2 = a^2 \left( 1 - e^2 \right)\]
\[ \Rightarrow b^2 = 25\left( 1 - \frac{16}{25} \right)\]
\[ \Rightarrow b^2 = 9\]
\[ \therefore \frac{x^2}{25} + \frac{y^2}{9} = 1\]
\[\text{ This is the required equation of the ellipse } .\]