$\mathrm{S}$ and $\mathrm{T}$ are the foci of an ellipse and $\mathrm{B}$ is an end of the minor axis. If STB is an equilateral triangle, then 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

Let eq. of ellipse be $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $\mathrm{S}$ is $(-\mathrm{ae}, 0), \mathrm{T}$ is $(\mathrm{ae}, 0)$ and $\mathrm{B}$ is $(0, \mathrm{~b})$
$\Rightarrow \mathrm{SB}=\sqrt{(0+\mathrm{ae})^{2}+\mathrm{b}^{2}}$ Also $\mathrm{SB}^{2}=\mathrm{ST}^{2} \Rightarrow 4 \mathrm{a}^{2} \mathrm{e}^{2}=\mathrm{a}^{2} \mathrm{e}^{2}+\mathrm{b}^{2}$ $\Rightarrow 3 a^{2} e^{2}=a^{2}\left(1-e^{2}\right)=a^{2}-a^{2} e^{2}$ $\Rightarrow 4 a^{2} e^{2}=a^{2} \Rightarrow e^{2}=\frac{1}{4} \Rightarrow e=\frac{1}{2}$

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