KVPY SX SB Exam 2018 Question Paper

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