Question

Write the eccentricity of an ellipse whose latus-rectum is one half of the minor axis.

Answer 1

Given: Length of latus-rectum is equal to half of minor axis

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

$\Rightarrow \frac{b}{a}=\frac{1}{2}[\because b\ne 0]$

$\Rightarrow \frac{b^2}{a^2}=\frac{1}{4}$

$\Rightarrow 1-\frac{b^2}{a^2}=\frac{3}{4}$

$\Rightarrow e^2=\frac{3}{4}$

$\therefore e=\frac{\sqrt{3}}{2}[\because e>0]$

Hence, the eccentricity is $e=\frac{\sqrt{3}}{2}$