Question

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

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

Answer 1

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


According to the question, the latus rectum of an ellipse is half its minor axis.
i.e. $\frac{2b^2}{a}=\frac{1}{2}\times 2b$
$\Rightarrow 2b^2=ab$
$\Rightarrow a=2b$
Now, $e=\sqrt{1-\frac{a^2}{b^2}}$
$\Rightarrow e=\sqrt{1-\frac{b^2}{4b^2}}$
$\Rightarrow e=\sqrt{1-\frac{1}{4}}$
$\Rightarrow e=\sqrt{\frac{3}{4}}$
$\Rightarrow e=\frac{\sqrt{3}}{2}$