An ellipse is inscribed in a circle and a point is inside a circle is choosen at random. If the

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10-2. Parabola, Ellipse, Hyperbola

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An ellipse is inscribed in a circle and a point is inside a circle is choosen at random. If the probability that this point lies outside the ellipse is $\frac {2}{3}$ then eccentricity of ellipse is $\frac{{a\sqrt b }}{c}$ . Where $gcd( a, c) = 1$ and $b$ is square free integer ($b$ is not divisible by square of any integer except $1$ ) then $a · b · c$ is

A

$11$

B

$12$

C

$16$

D

$18$

Solution

Radius of circle $=$ semi-major axis

$p=\frac{\pi\left(a^{2}-a b\right)}{\pi a^{2}}=1-\frac{b}{a}=\frac{2}{3}$

$\Rightarrow \frac{b}{a}=\frac{1}{3}$

$\Rightarrow 1-\frac{b^{2}}{a^{2}}=\frac{8}{9}$

$\Rightarrow \mathrm{e}=\frac{2 \sqrt{2}}{3}$

So, $a \cdot b \cdot c=2 \cdot 2 \cdot 3=12$

Standard 11

Mathematics

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