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Q.

Consider the circle $x^{2} + y^{2} = 4 a^{2}$ and the parabola $y^{2} = 3 a x,$ where a is the outcome of a fair die thrown once. What is the probability that the area of the smaller region bounded by the two curves is less than 100 square units ?

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a

$\frac{5}{6}$

b

$\frac{2}{3}$

c

$\frac{1}{2}$

d

$\frac{1}{3}$

answer is C.

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Detailed Solution

Point of intersection $(a, \pm \sqrt{3} a) .$ So area of the region

$= 2 \int_{0}^{a} \sqrt{3 a x} d x + 2 \int_{a}^{2 a} \sqrt{4 a^{2} - x^{2}} d x \begin{array}{l} = 2 \sqrt{3 a} {[\frac{x^{3 / 2}}{3 / 2}]}_{0}^{a} + 2 {[\frac{x}{2} \sqrt{4 a^{2} - x^{2}} + \frac{4 a^{2}}{2} \sin^{- 1} \frac{x}{2 a}]}_{a}^{2 a} \\ = (\frac{1}{\sqrt{3}} + \frac{4 π}{3}) a^{2} < 100 \\ \Rightarrow a \in {1, 2, 3, 4} \end{array}$

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