Question

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

• A. $\frac{1}{2}$
• B. $\frac{\sqrt{3}}{2}$
• C. $\frac{1}{\sqrt{3}}$
• D. none of these

Answer 1

The correct option is B $\frac{\sqrt{3}}{2}$

Let the equation of the ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and let $e$ be its eccentricity.

Latusrectum = $\frac{1}{2}$ (Minor Axis)

$\frac{2b^2}{a}=\frac{1}{2}\left(2b\right)$

$2b=a$

$4b^2=a^2$

$4a^2(1-e^2)=a^2$

$4-4e^2=1$

$4e^2=3$

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