Mathematics - Ellipse Question with Solution | TestHub

Given equations of ellipses $$\begin{array}{rl}& {\mathrm{E}}_1:\frac{x^2}{3}+\frac{y^2}{2}=1\\ & \Rightarrow e_1=\sqrt{1-\frac{2}{3}}=\frac{1}{\sqrt{3}}\\ & \text{ and }{\mathrm{E}}_2:\frac{x^2}{16}+\frac{y^2}{b^2}=1\\ & \Rightarrow e_2=\sqrt{\frac{1-b^2}{16}}=\sqrt{\frac{16-b^2}{4}}\\ & \text{ Also, given }{e}_1\times e_2=\frac{1}{2}\\ & \Rightarrow \frac{1}{\sqrt{3}}\times \sqrt{\frac{16-b^2}{4}}=\frac{1}{2}\Rightarrow 16-b^2=12\\ & \Rightarrow b^2=4\\ & \therefore \text{ Length of minor axis of }\\ & {\mathrm{E}}_2=2b=2\times 2=4\end{array}$$