Question

The difference between the lengths of the major axis and the latus-rectum of an ellipse is
(a) ae
(b) 2ae
(c) ae²
(d) 2ae²

Answer 1

$$\begin{gathered} \left(\mathrm{d}\right)2\mathrm{a}{\mathrm{e}}^2 \\ \mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{latus}\mathrm{rectum}=\frac{2b^2}{a} \\ \mathrm{and}e=\sqrt{1-\frac{b^2}{a^2}} \\ {a}^2{e}^2=a^2-b^2 \\ \Rightarrow b^2=a^2-a^2{e}^2 \\ \Rightarrow b^2=a^2\left(1-e^2\right) \\ \therefore \mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{latus}\mathrm{rectum}=\frac{2a^2\left(1-e^2\right)}{a}={\mathrm{2a(1-e}}^2\text{)} \\ \text{Length of the major axis=2a} \\ \mathrm{Difference}\mathrm{between}\mathrm{length}\mathrm{of}\mathrm{latus}\mathrm{rectum}\mathrm{and}\mathrm{length}\mathrm{of}\mathrm{major}\mathrm{axis}=2a-{\mathrm{2a(1-e}}^2\text{)} \\ =2a-2a+2a{e}^2 \\ =2a{e}^2 \end{gathered}$$