Question

If the minor axis of an ellipse subtends an angle ${60}^o$ at each focus then the eccentricity of then ellipse is:

• A. $\frac{1}{2}$
• B. $\sqrt{\frac{1}{2}}$
• C. $\sqrt{\frac{3}{2}}$
• D. None of these

Answer 1

The correct option is D None of these

slope of $AF=\frac{b}{-ae}$ ($m_1$)

slope of $BF=\frac{-b}{-ae}=\frac{b}{ae}$ $(${m}_{2}$)

$\tan {60}^o=\frac{m_1-m_2}{1+m_1{m}_2}=\frac{\frac{-2b}{ae}}{1-\frac{b^2}{a^2{e}^2}}$

$\sqrt{3}=\frac{-2abe}{a^2{e}^2-b^2}$

$=\frac{2e\sqrt{1-e^2}}{1-2e^2}$

check options $e\ne \frac{1}{2},e\ne \sqrt{\frac{1}{2}},e\ne \sqrt{\frac{3}{2}}$ option d is correct