Question

Let $S$ and $S^{\prime }$ be two foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ . A circle is inscribed in the ellipse. If $S{S}^{\prime }$ is the diameter of the circle, then the eccentricity $e$ of the ellipse is

• A. $\pm \frac{1}{\sqrt{2}}$
• B. $\frac{1}{3}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{1}{4}$

Answer 1

The correct option is C $\frac{1}{\sqrt{2}}$


The radius of the circle having $S{S}^{\prime }$ as diameter is $r=ae$.
It touches the ellipse then
$r=ae=b\Rightarrow e^2=\frac{b^2}{a^2}\Rightarrow e^2=1-e^2\Rightarrow e=\frac{1}{\sqrt{2}}$