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  4. An ellipse of eccentricity (2 √2/3) is inscribed in a circle. A point is chosen inside the circle at random. The probability that the point lies outside the ellipse is

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Q. An ellipse of eccentricity $\frac{2 \sqrt{2}}{3}$ is inscribed in a circle. A point is chosen inside the circle at random. The probability that the point lies outside the ellipse is

KEAMKEAM 2018

Solution:

Given, $e=\frac{2 \sqrt{2}}{3}$
$e^{2}=1-\frac{b^{2}}{a^{2}}$
$\Rightarrow \frac{b^{2}}{a^{2}}=1-\frac{8}{9} \,...(i)$
image
$P(x) =\frac{\pi a^{2}-\pi a b}{\pi a^{2}}$
$=1-\frac{b}{a}$
$=1-\frac{1}{3}$ [using Eq. (i)]
$P(x) =\frac{2}{3}$