The area (in sq. units) of the region $S=\{z \in \mathbb{C}:|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2, \operatorname{Im}(z) \geq 0\}$ is

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Aicharya · Accepted Answer

Put $z=x+i y$ $|z-1| \leq 2 \Rightarrow(x-1)^2+y^2 \leq 4$ ...(i) $\Rightarrow x-y \leq 1$ ...(ii) $\operatorname{Im}(\mathrm{z}) \geq 0 \Rightarrow \mathrm{y} \geq 0$ ...(iii)

$\begin{aligned} & \text { Required area } \\ & =\text { Area of semi-circle }- \text { area of sector A } \\ & \frac{1}{2} \pi(2)^2-\frac{\pi}{2} \\ & =\frac{3 \pi}{2}\end{aligned}$

The area (in sq. units) of the region $S=\{z \in \mathbb{C}:|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2,…

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