Question

If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is :

• A. $\frac{1}{2}$
• B. $\frac{2\sqrt{2}-1}{2}$
• C. $\sqrt{2}-1$
• D. $\frac{\sqrt{2}-1}{2}$

Answer 1

Answer: (C)

$S_{1} S_{2}=$ distance between foci
$=2ae=\frac{1}{2}\left(\right.$ length of latus rectum $\left.=\frac{2 b^{2}}{a}\right)$
$\therefore 2ae=\frac{1}{2}\left(\frac{2 b^{2}}{a}\right)$
$\Rightarrow 2 a^{2} e=b^{2}$
$\Rightarrow e=\frac{b^{2}}{2 a^{2}}=\frac{a^{2}\left(1-e^{2}\right)}{2 a^{2}}=\frac{1-e^{2}}{2}$
$\Rightarrow e^{2}+2 e-1=0$
$\Rightarrow e=\frac{-2+\sqrt{4+4}}{2}=\frac{\sqrt{8-2}}{2}=\sqrt{2}-1$