Question

If the minor axis of an ellipse subtends an angle 60° at each focus then the eccentricity of the ellipse is

• A. $\frac{\sqrt{3}}{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{2}{\sqrt{3}}$
• D. none

Answer 1

The correct option is A $\frac{\sqrt{3}}{2}$

From the right angled triangle, BOS,
$\tan {30}^{\circ }=\frac{OB}{OS}\Rightarrow \frac{1}{\sqrt{3}}=\frac{b}{ae}\Rightarrow a^2{e}^2=3b^2$
$\Rightarrow a^2{e}^2=3a^2(1-e^2)$
$\Rightarrow 4e^2=3\Rightarrow e=\frac{\sqrt{3}}{2}$