Question

If the length of the latus rectum of an ellipse is $4$ units and the distance between a focus and its nearest vertex on the major axis is $\frac{3}{2}$ units, then its eccentricity is :

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{2}{3}$
• D. $\frac{1}{9}$

Answer 1

The Correct Option is B

Given: $ \frac{2 b^2}{a}=4$
$\Rightarrow b^{2}=2 a$
$b^{2}=a^{2}\left(1-e^{2}\right)$
$a(1-e)=\frac{3}{2}$
So $2=a(1-e)(1+e) \Rightarrow 2=\frac{3}{2}(1+e)$
$ \Rightarrow 4=3+3 e$
$ \Rightarrow e=\frac{1}{3}$