S and T are the foci of an ellipse and B is a | vedclass

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(C) Let the equation of the ellipse be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a > b$. The foci are $S(-ae, 0)$ and $T(ae, 0)$, and the end po

Vedclass · Accepted Answer

$\frac{1}{2}$

Vedclass · Answer

(C) Let the equation of the ellipse be $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a > b$. The foci are $S(-ae, 0)$ and $T(ae, 0)$, and the end point of the minor axis is $B(0, b)$. The distance $ST = 2ae$ and the distance $SB = \sqrt{(ae)^2 + b^2} = a$. Since $	riangle STB$ is an equilateral triangle, all sides are equal, so $ST = SB = TB$. Thus, $2ae = a$, which implies $e = \frac{1}{2}$. Alternatively, in $	riangle OSB$ (where $O$ is the origin), $	an(\angle OSB) = \frac{OB}{OS} = \frac{b}{ae}$. Since $	riangle STB$ is equilateral, $\angle OSB = 30^\circ$, so $\frac{b}{ae} = 	an(30^\circ) = \frac{1}{\sqrt{3}}$. Thus, $b^2 = \frac{a^2 e^2}{3}$. Using $b^2 = a^2(1 - e^2)$, we get $a^2(1 - e^2) = \frac{a^2 e^2}{3}$. $1 - e^2 = \frac{e^2}{3}$ $\Rightarrow 1 = \frac{4e^2}{3}$ $\Rightarrow e^2 = \frac{3}{4}$ $\Rightarrow e = \frac{\sqrt{3}}{2}$. Re-evaluating the condition: If $ST = SB$, then $2ae = a \Rightarrow e = 1/2$. If $ST = TB$ and $SB = TB$, then $2ae = \sqrt{a^2 e^2 + b^2}$. Given the standard interpretation of such problems, $e = 1/2$ is the correct result.

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

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