Question

The eccentricity of the ellipse \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] if its latus rectum is equal to one half of its minor axis, is

 

Options

• \[\frac{1}{\sqrt{2}}\]

   

• \[\frac{\sqrt{3}}{2}\]

   

• \[\frac{1}{2}\]

   

• none of these

Answer 1

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