Sand $T$ are the foci of an ellipse and $B$ is the end point of the minor axis. If $S T B$ is equilateral triangle, the eccentricity of the ellipse is

• A

  $\frac{1}{4}$
• B

  $\frac{1}{3}$
• Correct answer
  C

  $\frac{1}{2}$
• D

  $\frac{2}{3}$

Solution

Equation of the ellipse is $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $$ Foci are $S(a e, 0), T(-a e, 0)$ $B(0, b)$ is the end of the minor axis. STB is an equilateral triangle. $$ \begin{aligned} S B &=S T \\ S B^{2} &=S T^{2} \end{aligned} $$ $\Rightarrow \quad a^{2} e^{2}+b^{2}=4 a^{2} e^{2}$ $\Rightarrow \quad b^{2}=3 a^{2} e^{2}$ $\Rightarrow \quad a^{2}\left(1-e^{2}\right)=3 a^{2} e^{2}$ $\Rightarrow \quad 1-e^{2}=3 e^{2}$ $\Rightarrow \quad 4 e^{2}=1$ $\Rightarrow \quad e^{2}=\frac{1}{4}$ $\Rightarrow \quad e=\frac{1}{2}$ Eccentricity of the ellipse, $e=\frac{1}{2}$

Hence, the correct answer is option: 'C'.