Question

A circle is inscribed in an ellipse. If p is the probability that a point within the ellipse chosen randomly lies outside the circle, then eccentricity of the ellipse is

• A. $\sqrt{1-p}$
• B. $\sqrt{1-p^2}$
• C. $\sqrt{1-{(1-P)}^2}$
• D. $\sqrt{{(1+P)}^2-1}$

Answer 1

The correct option is C $\sqrt{1-{(1-P)}^2}$

Referring to the illustration:

Area of the circle: $\pi a^2$

Area of the ellipse: $\pi ab$

Hence, we have: $\frac{(\pi ab-\pi a^2)}{\pi ab}=(1-\frac{a}{b})$

Thus, $p=(1-\frac{a}{b})=>\frac{a}{b}=1-p$

Eccentricity of an ellipse = $\sqrt{1-{\left(\frac{a}{b}\right)}^2}=\sqrt{1-{(1-p)}^2}$

Hence, (c) is correct.