question_answer

S and T are the foci of the ellipse \[\left( \frac{{{x}^{2}}}{{{a}^{2}}} \right)+\left( \frac{{{y}^{2}}}{{{b}^{2}}} \right)=1\]and B is an end minor axis. If STB is an equilateral triangle, the eccentricity of the ellipse is:

A)  \[\frac{1}{4}\]                  

B)         \[\frac{1}{3}\]

C)  \[\frac{1}{2}\]                  

D)         \[\sqrt{\frac{3}{2}}\]

E)  \[\frac{1}{\sqrt{2}}\]

Correct Answer: C

Solution :

In \[\Delta BOT,\] \[\frac{b}{ae}=\tan 60{}^\circ \Rightarrow b=ae\sqrt{3}\] \[\therefore \]  \[e=\sqrt{1-\frac{{{b}^{2}}}{{{a}^{2}}}}=\sqrt{1-\frac{{{a}^{2}}{{e}^{2}}3}{{{a}^{2}}}}\] \[\Rightarrow \]               \[{{e}^{2}}=1-3{{e}^{2}}\Rightarrow 4{{e}^{2}}=1\] \[\Rightarrow \]               \[{{e}^{2}}=\frac{1}{4}\] \[\Rightarrow \]               \[e=\pm \frac{1}{2}\]