$P$ and $Q$ are two points on the upper half of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2

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$P$ and $Q$ are two points on the upper half of the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$. The centre of the ellipse is at the origin ' $O$ ' and $PQ$ is parallel to the $x$-axis such that the triangle $OPQ$ has the maximum possible area. A point is randomly selected from inside of the upper half of the ellipse. The probability that is lies outside the triangle, is :

Accepted Answer

Sol.(a) Area of$$\begin{matrix} \bigtriangleup OPQ & \ = \frac{1}{2}(2acos	heta)(bsin	heta) \ & \ = \frac{ab}{2}sin2	heta \leq \frac{ab}{2} \end{matrix}$$$$	ext{~}	ext{Probability}	ext{~} = \frac{\frac{\pi ab}{2} - \frac{ab}{2}}{\frac{\pi ab}{2}} = \frac{\pi - 1}{\pi}$$

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