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Question: An ellipse is inscribed in a circle and a point which in the circle is chosen at random. If the prob...

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

A

223\frac{2\sqrt{2}}{3}

B

53\frac{\sqrt{5}}{3}

C

89\frac{8}{9}

D

23\frac{2}{3}

Answer

223\frac{2\sqrt{2}}{3}

Explanation

Solution

23=πa2−πabπa2=1−ba=1−1−e2\frac{\mathbf{2}}{\mathbf{3}}\mathbf{=}\frac{\mathbf{\pi}\mathbf{a}^{\mathbf{2}}\mathbf{- \pi ab}}{\mathbf{\pi}\mathbf{a}^{\mathbf{2}}}\mathbf{= 1 -}\frac{\mathbf{b}}{\mathbf{a}}\mathbf{= 1 -}\sqrt{\mathbf{1 -}\mathbf{e}^{\mathbf{2}}}⇒ e2=89e^{2} = \frac{8}{9}

⇒ e=223e = \frac{2\sqrt{2}}{3}