Question

The length of the semi-latus rectum of an ellipse is one third of its major axis, its eccentricity would be

• A. $\frac{2}{3}$
• B. $\sqrt{\frac{2}{3}}$
• C. $\frac{1}{\sqrt{3}}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

Answer: (C)

Let eq. of ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$,
length $=\frac{b^2}{a}=\frac{a^2\left(1-e^2\right)}{a}=a\left(1-e^2\right)$
Given $a\left(1-e^2\right)=\frac{1}{3}(2a)$
$\Rightarrow 1-e^2=\frac{2}{3}$
$\Rightarrow e^2=1-\frac{2}{3}=\frac{1}{3}$
$\Rightarrow e=\frac{1}{\sqrt{3}}$