Question

If the distance between the foci of an ellipse is equal to the length of the latus-rectum, write the eccentricity of the ellipse.

Answer 1

\[\text{ According to the question, the distance between the foci of an ellipse is equal to the length of the latus rectum }.\]
\[i . e . \frac{2 b^2}{a} = 2ae\]
\[ \Rightarrow e = \frac{b^2}{a^2}\]
\[\text{ But } e = \sqrt{1 - \frac{b^2}{a^2}}\]
\[\text{ Then } e = \sqrt{1 - e} \]
\[\text{ Squaring both sides, we get }:\]
\[ e^2 + e - 1 = 0\]
\[ \Rightarrow e = \frac{- 1 \pm \sqrt{1 + 4}}{2} \left( \because \text{ Ecentricity cannot be negative } \right)\]
\[ \Rightarrow e = \frac{\sqrt{5} - 1}{2}\]