Question

Let $S=S_1\cap S_2\cap S_3,$ where
$S_1=\{zϵC:|z|<4\},S_2=\{zϵC:Im\left[\frac{z-1+\sqrt{3}i}{1-\sqrt{3}i}\right]>0\}and{S}_3=\{zϵC:Re(z)>0\}.$
Area of $S=$

• A. $\frac{10\pi }{3}$
• B. $\frac{20\pi }{3}$
• C. $\frac{16\pi }{3}$
• D. $\frac{32\pi }{3}$

Answer 1

The correct option is B $\frac{20\pi }{3}$

$S_1:x^2+y^2<16S_2:Im\left[\frac{z-1+\sqrt{3}i}{1-\sqrt{3}i}\right]>0\Rightarrow \sqrt{3}(x-1)+(y+\sqrt{3})>0\Rightarrow y+\sqrt{3x}>0S_3:x>0$
Then $S:S_1\cap S_2\cap S_3$ is as shown in the figure given below.

Area of shaded region
$=\frac{\pi }{4}\times 4^2+\frac{\pi \times 4^2\times {60}^{\circ }}{{360}^{\circ }}=4\pi +\frac{8\pi }{3}=\frac{20\pi }{3}$