An ellipse is inscribed in a circle and a point within circle is chosen at random. If probability th

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10-2. Parabola, Ellipse, Hyperbola

normal

An ellipse is inscribed in a circle and a point within the circle is chosen at random. If the probability that this point lies outside the ellipse is $2/3 $ then the eccentricity of the ellipse is :

$\frac{{2\sqrt 2 }}{3}$

$\frac{{\sqrt 5 }}{3}$

$\frac{8}{9}$

$\frac{2}{3}$

Solution

$\frac{2}{3}=\frac{{\pi \,{a^2}\,\, – \,\,\pi \,ab}}{{\pi \,{a^2}}} = 1 – \frac{b}{a} = 1 – \sqrt {1\,\, – \,\,{e^2}} $
$\Rightarrow $ $e^2 =\frac{8}{9}$ $ \Rightarrow$ $ e =\frac{{2\sqrt 2 }}{3}$

Standard 11

Mathematics

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easy

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normal

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hard

(JEE MAIN-2023)

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hard

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medium