Question

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

Answer 1

For ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, length of latus-rectum is $\frac{2b^2}{a}$

Condition:Latus-rectum is one half of the minor axis.

$\therefore \frac{2b^2}{a}=\frac{1}{2}2b$

$\therefore \frac{b}{a}=\frac{1}{2}$

Eccentricity is given by,

$e=\sqrt{1-\frac{b^2}{a^2}}$

$e=\sqrt{1-\frac{1}{4}}$

$e=\sqrt{\frac{3}{4}}$

$e=\frac{\sqrt{3}}{2}$