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

$$\begin{gathered} \left(\mathrm{c}\right)\frac{\sqrt{3}}{2} \\ \text{According to the question, the latus rectum of an ellipse is half its minor axis.} \\ \mathrm{i}.\mathrm{e}.\frac{2b^2}{a}=\frac{1}{2}\times 2b \\ \Rightarrow 2b^2=ab \\ \Rightarrow a=2b \\ \mathrm{Now},e=\sqrt{1-\frac{b^2}{a^2}} \\ \Rightarrow e=\sqrt{1-\frac{b^2}{4b^2}} \\ \Rightarrow e=\sqrt{1-\frac{1}{4}} \\ \Rightarrow e=\sqrt{\frac{3}{4}} \\ \Rightarrow e=\frac{\sqrt{3}}{2} \end{gathered}$$