Question

If the points of intersection of the circle $x^2+y^2=16$ with $x-\text{axis}$ are foci of an ellipse and points of intersection of circle $x^2+y^2=16$ with $y-\text{axis}$ are end points of minor axis of the same ellipse, then eccentricity of the ellipse is

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

Answer 1

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

Points of intersection of the circle with $x-\text{axis}$ are foci of an ellipse
$\Rightarrow ae=4\cdots \left(i\right)$
Also, the points of intersection of circle with $y-\text{axis}$ are end points of minor axis of the same ellipse
$\Rightarrow b=4\cdots \left(ii\right)$
From $\left(i\right)$ and $\left(ii\right)$
$ae=b\Rightarrow a^2{e}^2=b^2=a^2-a^2{e}^2\Rightarrow e^2=1-e^2(\because a^2\ne 0)\therefore e=\frac{1}{\sqrt{2}}$