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

Answer 1

The correct option is D $\frac{1}{3}$

$\text{Length of L.R.}=\frac{2b^2}{a}=4\Rightarrow b^2=2a\text{Distance between focus and vertex}=\frac{3}{2}\Rightarrow \sqrt{(a-ae{)}^2+0}=\frac{3}{2}\Rightarrow a(1-e)=\frac{3}{2}......\left(i\right)\text{Eccentricity,}e=\frac{\sqrt{a^2-b^2}}{a}\Rightarrow b^2=a^2(1-e^2)\Rightarrow 2a=a.a(1+e)(1-e)\Rightarrow 2=\frac{3}{2}(1+e)\text{(from (i))}\Rightarrow e=\frac{1}{3}$