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10-1.Circle and System of Circles
normal
In the given figure, $AB$ is tangent to the circle with centre $O$ , the ratio of the shaded region to the unshaded region of the triangle $OAB$ is
A
$\frac{{2\sqrt 3 - 2}}{\pi }$
B
$\frac{{3\sqrt 3 - 2}}{\pi }$
C
$\frac{{2 - \sqrt 3 }}{\pi }$
D
$\frac{{3\sqrt 3 }}{\pi } - 1$
Solution
In $\Delta \mathrm{AOB}, \mathrm{AB}=2 \tan 60^{\circ}=2 \sqrt{3}$
$\Rightarrow$ Area of $\Delta \mathrm{AOB}=\frac{1}{2} \times 2 \times 2 \sqrt{3}=2 \sqrt{3}$
Area of sector $\mathrm{O} \mathrm{AC}=\frac{60}{360} \pi(2)^{2}=\frac{2 \pi}{3}$
$\Rightarrow$ Ratio $=\frac{2 \sqrt{3}-\frac{2 \pi}{3}}{\frac{2 \pi}{3}}=\frac{3 \sqrt{3}}{\pi}-1$
Standard 11
Mathematics
