Question

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

Answer 1

According to given,$\frac{2b^2}{a}=b\Rightarrow 2b=a$

$\Rightarrow b=\frac{a}{2}$

We know that $c^2=a^2-b^2=a^2-\frac{a^2}{4}=\frac{4a^2-a^2}{4}=\frac{3a^2}{4}$

$\Rightarrow \frac{c^2}{a^2}=\frac{3}{4}$

$\Rightarrow \frac{c}{a}=\frac{\sqrt{3}}{2}$

Hence, the eccentricity of the ellipse$=\frac{\sqrt{3}}{2}$