Jee Mains
The area (in sq. units) of the region $\mathrm{S}=\{\mathrm{z} \in \mathrm{C} ;|\mathrm{z}-1| \leq 2 ;(\mathrm{z}+\overline{\mathrm{z}})+\mathrm{i}(\mathrm{z}-\overline{\mathrm{z}}) \leq 2, \operatorname{Im}(\mathrm{z}) \geq 0\}$ is
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Solution
According to the question, Put $\mathrm{z}=\mathrm{x}+\mathrm{iy} \Rightarrow(\mathrm{z}+\overline{\mathrm{z}})+\mathrm{i}(\mathrm{z}-\overline{\mathrm{z}}) \leq 2 \Rightarrow 2 \mathrm{x}+\mathrm{i}(2 \mathrm{iy}) \leq 2$
\[ \begin{aligned} & \Rightarrow \mathrm{x}-\mathrm{y} \leq 1 \\ & |\mathrm{z}-1| \leq 2 \end{aligned} \]interior part of circle with centre $(1,0)$ and $\mathrm{r}=2$
\[ \operatorname{Im}(\mathrm{z}) \geq 0 \Rightarrow \mathrm{y} \geq 0 \]Required area = Area of semi-circle-area of sector $\mathrm{A}=\frac{1}{2} \pi(2)^{2}-\frac{\pi^{2}}{2}\left(\frac{\pi}{4}\right)=\frac{3 \pi}{2}$
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