Explanation

(b)
Given ellipse is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ whose are is$\ \pi\ ab$. The auxiliary circle to the given ellipse is $x^{2} + y^{2} = a^{2}$ whose area is $\pi a^{2}$
Given that, $\pi a^{2} = 2\pi ab \Rightarrow \ a = 2b\$
Now, eccentricity of ellipse
$= \sqrt{1 - \frac{b^{2}}{a^{2}}}$
$= \sqrt{1 - \frac{b^{2}}{4b^{2}}\ } = \frac{\sqrt{3}}{2}$