Question

If the latus rectum of an ellipse is equal to half of its minor axis, then its eccentricity is

Answer 1

Let the equation of ellipse be
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b)$
Length of latus rectum $=\frac{2b^2}{a},$
and length of minor axis $=2b.$

Given,$\frac{2b^2}{a}=b$
$\Rightarrow \frac{b}{a}=\frac{1}{2}(\because b\ne 0)$
$\Rightarrow \frac{b^2}{a^2}=\frac{1}{4}$
$\therefore e=\sqrt{1-\frac{b^2}{a^2}}=\frac{\sqrt{3}}{2}$