Question
Write the eccentricity of an ellipse whose latus-rectum is one half of the minor axis.

Answer

According to the question, the latus rectum is half its minor axis. i.e. $\frac{2\text{b}^2}{\text{a}}=\frac{1}{2}\times2\text{b}$ $\Rightarrow2\text{b}=\text{a}$ Now, $\text{e}=\sqrt{1-\frac{\text{b}^2}{\text{a}^2}}$ $\Rightarrow\text{e}=\sqrt{1-\frac{\text{b}^2}{4\text{b}^2}}$ $\Rightarrow\text{e}=\sqrt{1-\frac{1}{4}}$ $\Rightarrow\text{e}=\frac{\sqrt{3}}{2}$

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