Question

The eccentricity of the ellipse, if the distance between the foci is equal to the length of the latus-rectum, is
(a) $\frac{\sqrt{5}-1}{2}$

(b) $\frac{\sqrt{5}+1}{2}$

(c) $\frac{\sqrt{5}-1}{4}$

(d) none of these

Answer 1

$$\begin{gathered} \left(\mathrm{a}\right)e=\frac{\sqrt{5}-1}{2} \\ \text{According to the question, the distance between the foci is equal to the length of the latus rectum.} \\ \frac{2b^2}{a}=2ae \\ \Rightarrow b^2=a^{2}e \\ \mathrm{Now},e=\sqrt{1-\frac{b^2}{a^2}} \\ \Rightarrow e=\sqrt{1-\frac{a^{2}e}{a^2}} \\ \Rightarrow e=\sqrt{1-e} \\ \text{On squaring both sides, we get:} \\ {e}^2+e-1=0 \\ \Rightarrow e=\frac{-1\pm \sqrt{1+4}}{2} \\ \Rightarrow e=\frac{\sqrt{5}-1}{2}\left(\because e\mathrm{cannot}\mathrm{be}\mathrm{negative}\right) \end{gathered}$$