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10-2. Parabola, Ellipse, Hyperbola
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Let $S = 0$ is an ellipse whose vartices are the extremities of minor axis of the ellipse $E:\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1,a > b$ If $S = 0$ passes through the foci of $E$ , then its eccentricity is (considering the eccentricity of $E$ as $e$ )
A
$\sqrt {\frac{{1 - 2{e^2}}}{{1 - {e^2}}}} $
B
$\frac{1}{{\sqrt {1 + {e^2}} }}$
C
$\frac{{1 - 2{e^2}}}{{1 - {e^2}}}$
D
$\frac{{{e^2}}}{{1 + {e^2}}}$
Solution
$\frac{b^{2}}{a^{2}}=1-e^{2}$ and $a^{2} e^{2}=b^{2}\left(1-e_{1}^{2}\right)$
$\Rightarrow 1-\mathrm{e}_{1}^{2}=\frac{\mathrm{e}^{2}}{1-\mathrm{e}^{2}}$
$ \Rightarrow \mathrm{e}_{1}^{2}=1-\frac{\mathrm{e}^{2}}{1-\mathrm{e}^{2}}$
$e_{1}=\sqrt{\frac{1-2 e^{2}}{1-e^{2}}}$
Standard 11
Mathematics
