Question

If the latus-rectum of an ellipse is one half of its minor axis, then its eccentricity is

• A. $\frac{1}{2}$
• B. $\frac{\sqrt{3}}{4}$
  
  
• C. $\frac{\sqrt{3}}{2}$
  
• D. $\frac{1}{\sqrt{2}}$
  

Answer 1

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

Given: Length of latus-rectum is equal to half of minor axis

$\Rightarrow \frac{2b^2}{a}=b$

$\Rightarrow \frac{b}{a}=\frac{1}{2}[\because b\ne 0]$

$\Rightarrow \frac{b^2}{a^2}=\frac{1}{4}$

$\Rightarrow 1-\frac{b^2}{a^2}=\frac{3}{4}$

$\Rightarrow e^2=\frac{3}{4}$

$\therefore e=\frac{\sqrt{3}}{2}[\because e>0]$

Hence, option $\left(C\right)$ is correct.