Question

$S$ and $T$ are the foci of an ellipse and $B$ 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. $\frac{1}{3}$
• C. $\frac{1}{2}$
• D. $\frac{2}{3}$

Answer 1

Answer: (C)

$\frac{1}{2}$

Answer 2

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