Probability + Ellipse

Probability Level 3

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 \large \dfrac{2}{3} , then the eccentricity of the ellipse is a b c \large \frac{a\sqrt b} {c} where gcd ⁡ ( a , c ) = 1 \gcd(a,c)=1 and b b is a square free integer . Then a × b × c \large{a\times b\times c} is

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The answer is 12.

Akshay Sharma by Akshay Sharma , 21, India

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1 solution

The radius of circle = Semi-Major axis of ellipse. Let this be a, and the semi minor axis be b. Eccentricity of the ellipse = e.

The given probability = π ( a 2 − a b ) π a 2 = 1 − b a \dfrac{\pi(a^{2}-ab)}{\pi a^{2}} = 1 - \dfrac{b}{a}
1 − b a = 2 3 1 - \dfrac{b}{a} = \dfrac{2}{3}
b a = 1 3 \dfrac{b}{a} = \dfrac{1}{3}
1 − e 2 = 1 3 \sqrt{1-e^{2}} = \dfrac{1}{3}
e = 2 2 3 e = \dfrac{2\sqrt{2}}{3}
a b c = 2 ⋅ 2 ⋅ 3 = 12 abc = 2 \cdot 2 \cdot 3 = 12


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