Question

In an ellipse, if the lines joining a focus to the extremities of the minor axis make an equilateral triangle with the minor axis, the eccentricity of the ellipse is

• A. $3/4$
• B. $\sqrt{3}/2$
• C. $1/2$
• D. $2/3$

Answer 1

The correct option is B $\sqrt{3}/2$

Let one of the focus of the ellipse be $S(ae,0)$ and the minor axes are $B(0,b)$ and $B^{\prime }(0,-b)$

Given triangle $SB{B}^{\prime }$ is equilateral

$\Rightarrow SB=B{B}^{\prime }$

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

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

$\therefore e=\frac{\sqrt{3}}{2}$