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

Answer 1

The correct option is B

$\frac{\sqrt{3}}{2}$

Ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ focus (ae, 0), extremities of minor axes (0, b), (0, -b).
$(ae{)}^2+b^2=(2b{)}^2\Rightarrow 3b^2=a^2{e}^2\Rightarrow 3a^2(1-e^2)=a^2{e}^2\Rightarrow e^2=\frac{3}{4}\Rightarrow e=\frac{\sqrt{3}}{2}$