Find the equation of the ellipse in the standard form whose minor axis is equal to the distance between foci and whose latus-rectum is 10.

#### Solution

\[\text{ According to the question, the minor axis is equal to the distance between the foci }.\]

\[\text{ i . e } . 2b = 2\text{ ae and } \frac{{2b}^2}{a} = 10 \text{ or } b^2 = 5a \]

\[ \Rightarrow b = ae\]

\[ \Rightarrow b^2 = a^2 e^2 \]

\[ \Rightarrow b^2 = a^2 \left( 1 - \frac{b^2}{a^2} \right) \left( \because e = \sqrt{1 - \frac{b^2}{a^2}} \right)\]

\[ \Rightarrow b^2 = a^2 - b^2 \]

\[ \Rightarrow a^2 = 2 b^2 \]

\[ \Rightarrow a^2 = 10a \left( \because b^2 = 5a \right)\]

\[ \Rightarrow a = 10\]

\[ \Rightarrow b^2 = 5a \]

\[ \Rightarrow b^2 = 50\]

\[\text{ Substituting the values ofaandbin the equation of an ellipse, we get }:\]

\[\frac{x^2}{100} + \frac{y^2}{50} = 1\]

\[ \therefore x^2 + 2 y^2 = 100\]

\[\text{This is the required equation of the ellipse }.\]