Question

A point is chosen inside the auxilary circle of an ellipse of eccentricity $\frac{2\sqrt{2}}{3}$ at random. The probability that the point lies outside the ellipse is

• A. $\frac{4}{9}$
• B. $\frac{2}{3}$
• C. $\frac{1}{3\sqrt{2}}$
• D. $\frac{1}{9}$

Answer 1

The correct option is B $\frac{2}{3}$

$e=\frac{2\sqrt{2}}{3}$
$e^2=1-\frac{b^2}{a^2}$
$\frac{b^2}{a^2}=1-\frac{8}{9}=\frac{1}{9}$
$P\left(\text{point lies outside}\right)=\frac{\pi a^2-\pi ab}{\pi a^2}=1-\frac{b}{a}=1-\frac{1}{3}=\frac{2}{3}$