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(B) Let the radius of the circle be $R = a$ (where $a$ is the semi-major axis of the ellipse). The area of the circle is $A_c = \pi a^2$. The area of

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$12$

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(B) Let the radius of the circle be $R = a$ (where $a$ is the semi-major axis of the ellipse). The area of the circle is $A_c = \pi a^2$. The area of the inscribed ellipse with semi-major axis $a$ and semi-minor axis $b'$ is $A_e = \pi a b'$. The probability that a point chosen at random inside the circle lies outside the ellipse is given by $P = 1 - \frac{A_e}{A_c} = 1 - \frac{\pi a b'}{\pi a^2} = 1 - \frac{b'}{a} = \frac{2}{3}$. This implies $\frac{b'}{a} = \frac{1}{3}$,so $b' = \frac{a}{3}$. The eccentricity $e$ of the ellipse is given by $e = \sqrt{1 - \frac{(b')^2}{a^2}} = \sqrt{1 - (\frac{1}{3})^2} = \sqrt{1 - \frac{1}{9}} = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3}$. Comparing this with $\frac{a\sqrt{b}}{c}$,we have $a=2, b=2, c=3$. Since $\gcd(2, 3) = 1$ and $2$ is square-free,the value is $a \cdot b \cdot c = 2 \cdot 2 \cdot 3 = 12$.

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