Question

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 $\frac{2}{3}$ then the eccentricity of the ellipse is

• A. $\frac{2\sqrt{2}}{3}$
• B. $\sqrt{5}$
• C. $8$
• D. $2$

Answer 1

Answer: (A)

$\frac{2\sqrt{2}}{3}$

Probability $p=\frac{2}{3}$

Let the radius of circle$=a$

Major axis$=2a$

Minor axis$=2b$

Area of circles$=\pi a^2$

Area of ellipse$=\pi ab$

as because

$P=\frac{\pi ab-\pi a^2}{\pi ab}=1-\frac{a}{b}$

$\frac{a}{b}=1-\frac{2}{3}=\frac{1}{3}$

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

$=\sqrt{\frac{8}{9}}$

Eccentricity$=e=\frac{2\sqrt{2}}{3}$.