Question

Find the equation of the ellipse in the following case: 

Vertices (± 6, 0), foci (± 4, 0) 

Answer 1

\[\text{ Vertices }\left( \pm 6, 0 \right)\text{ and focus }\left( \pm 4, 0 \right)\]
\[\text{ The coordinates of its vertices and foci are }\left( \pm a, 0 \right)\text{ and }\left( \pm ae, 0 \right), \text{ respectively }.\]
\[i . e . , a = 6 \text{ and } ae = 4\]
\[ \therefore e = \frac{4}{6}\text{ or }\frac{2}{3}\]
\[\text{ Now }, b^2 = a^2 \left( 1 - e^2 \right)\]
\[ \Rightarrow b^2 = 36\left( 1 - \frac{16}{36} \right)\]
\[ \Rightarrow b^2 = 20\]
\[ \therefore \frac{x^2}{36} + \frac{y^2}{20} = 1\]
\[\text {This is the required equation of the ellipse }.\]