Question

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b$, be an ellipse with foci $F_1$ and $F_2$. Let $AO$ be its semi-minor axis, where $O$ is the centre of the ellipse. The lines $A{F}_1$ and $A{F}_2$, when extended, cut the ellipse again at points $B$ and $C$ respectively. Suppose that the triangle $ABC$ is equilateral. Then the eccentricity of the ellipse is

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

Answer 1

The correct option is D $\frac{1}{2}$


In $△OA{F}_2,$
$\frac{b}{ae}=\tan {60}^{\circ }=\sqrt{3}\Rightarrow \frac{b^2}{a^2}=3e^2\Rightarrow 1-e^2=3e^2[\because e^2=1-\frac{b^2}{a^2}]$
$\Rightarrow e^2=\frac{1}{4}\Rightarrow e=\frac{1}{2}$