Question

The eccentricity of the ellipse, if the minor axis is equal to the distance between the foci, is
(a) $\frac{\sqrt{3}}{2}$

(b) $\frac{2}{\sqrt{3}}$

(c) $\frac{1}{\sqrt{2}}$

(d) $\frac{\sqrt{2}}{3}$

Answer 1

$$\begin{gathered} \left(c\right)\frac{1}{\sqrt{2}} \\ \text{According to the question, the minor axis is equal to the distance between the foci.} \\ \mathrm{i}.\mathrm{e}.2b=2ae \\ \Rightarrow e=\frac{b}{a}...\left(1\right) \\ \mathrm{Now},e=\sqrt{1-\frac{b^2}{a^2}} \\ \Rightarrow e=\sqrt{1-e^2}[\mathrm{From}(1\left)\right] \\ \text{On squaring both sides, we get:} \\ {e}^2=1-e^2 \\ \Rightarrow 2e^2=1 \\ \Rightarrow e^2=\frac{1}{2} \\ \Rightarrow e=\frac{1}{\sqrt{2}} \end{gathered}$$