Question

If in any ellipse angle between lines joining the end points of minor axis and its foci is ${90}^o$, then find the eccentricity.

Answer 1

Let half lengths of major and minor axes be $a$ and $b$ respectively and let the major axis be along the x-axis.

Then one of the foci is $(ae,0)$ and the end points of the minor axis are $(0,b)$ and $(0,-b)$.

The other foci is $(-ae,0)$

Slopes of the lines joining the end points of the minor axes to the focus are:

$\frac{-b}{ae}$ and $\frac{b}{ae}$ respectively.

Since these lines are perpendicular, we must have:

$\frac{-b^2}{a^2{e}^2}=-1$

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

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

Thus the eccentricity of the ellipse is $\frac{1}{\sqrt{2}}$