Question

Find the eccentricity of an ellipse whose latus rectum is
(i) half of its minor axis
(ii) half of its major axis.

Answer 1

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