Mathematics - Complex Number Question with Solution | TestHub

$\begin{array}{rl}& \text{ Let }z=x+iy\\ & (x+iy)(1+i)+(x-iy)(1-i)=4\\ & x+ix+iy-y+x-ix-iy-y=4\\ & 2x-2y=4\\ & x-y=2\\ & |z-3|\le 1\\ & (x-3{)}^2+y^2\le 1\end{array}$ Area of shaded region $=\frac{\pi \cdot 1^2}{4}-\frac{1}{2}\cdot 1\cdot 1=\frac{\pi }{4}-\frac{1}{2}$ Area of unshaded region inside the circle $=\frac{3}{4}\pi \cdot 1^2+\frac{1}{2}\cdot 1\cdot 1=\frac{3\pi }{4}+\frac{1}{2}$ $\therefore$ difference of area $=\left(\frac{3\pi }{4}+\frac{1}{2}\right)-\left(\frac{\pi }{4}-\frac{1}{2}\right)$ $=\frac{\pi }{2}+1$