Question

A circle is described with the minor axis of an ellipse as a diameter. If the foci lie on the circle, then the eccentricity of the ellipse is

Answer 1

The length of the minor axis$=2b$

The distance between the foci$=2ae$

The line segment joining the foci is bisected perpendicular by the minor axis.

Hence the line segment joining the foci should also be a diameter. Hence they are equal.

$2b=2ae$

$b=ae$

$b^2=a^2{e}^2$

$a^2(1-e^2)=a^2{e}^2$

$1-e^2=e^2$

$1=2e^2$

$e^2=\frac{1}{2}$

$e=\frac{1}{\sqrt{2}}$

Hence the eccentricity $=\frac{1}{\sqrt{2}}$.