Question

If $S$ and $S^{\prime }$ are two foci of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $B$ is an end of the minor axis such that $\Delta BS{S}^{\prime }$ is equilateral, then write the eccentricity of the ellipse.

Answer 1

Diagram

In equilateral $\Delta BS{S}^{\prime },\angle B{S}^{\prime }S={60}^{\circ }$

$\Rightarrow \tan {60}^{\circ }=\frac{b}{ae}$

$\Rightarrow \sqrt{3}ae=b$

$\Rightarrow 3a^2{e}^2=b^2$

$\Rightarrow 3a^2{e}^2=a^2(1-e^2)$

$\Rightarrow 3e^2=1-e^2[\because a\ne 0]$

$\Rightarrow e^2=\frac{1}{4}$

$\Rightarrow e=\frac{1}{2}[\because e>0]$