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  • S and T are the foci of an ellipse and B is an end of the minor axis. If STB is an equilateral triangle, the eccentricity of the ellipse is  <div class

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

 

Option: 1

1/4

 


Option: 2

1/3


Option: 3

1/2


Option: 4

2/3


Answers (1)

best_answer

Let the equation of ellipse be   \mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1}

\mathrm{ \mathrm{S} \equiv(-\mathrm{ae}, 0) \\ }

\mathrm{ \mathrm{T} \equiv(\mathrm{ae}, 0) \\ }

\mathrm{ \mathrm{B} \equiv(0, \mathrm{~b}) \Rightarrow }

\mathrm{\mathrm{SB}=\sqrt{(0+\mathrm{ae})^2+\mathrm{b}^2} }
\mathrm{ \text { Also } \mathrm{SB}^2=\mathrm{ST}^2 \Rightarrow 4 \mathrm{a}^2 \mathrm{e}^2=\mathrm{a}^2 \mathrm{e}^2+\mathrm{b}^2 \\ }\mathrm{\text { or } \quad 3 \mathrm{a}^2 \mathrm{e}^2=\mathrm{a}^2\left(1-\mathrm{e}^2\right) \\ }
\mathrm{ =\mathrm{a}^2-\mathrm{a}^2 \mathrm{e}^2 \Rightarrow \mathrm{e}^2=1 / 4 \Rightarrow \mathrm{e}=1 / 2 }

Hence (C) is the correct answer.

Posted by

Irshad Anwar

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