The area (in sq.

Question

The area (in sq. units) of the region $S={z\in C:|z-1|\le 2;(z+\vec{z})+i(z-\vec{z})\le 2,lm(z)\ge 0}$ is

Accepted Answer

$$|z-1| \leq 2 \quad \Rightarrow \quad(x-1)^2+y^2=4$$

$$\begin{aligned}
& z+\bar{z}+i(z-\bar{z}) \leq 2 \
\Rightarrow \quad & x-y \leq 1 \
& \operatorname{Im}(z) \geq 0 \
\Rightarrow \quad & y \geq 0
\end{aligned}$$

$$\begin{aligned}
& \text { Required area }=\left(\frac{\frac{3 \pi}{4}}{2 \pi}\right)(\pi)(2)^2 \
& =\frac{3}{2} \pi
\end{aligned}$$

Solutions · 1