Question

Let $A,B,C$ be three sets of complex numbers as defined below:
$A=\{z:Im(z)\ge 1\}B=\{z:|z-2-i|=3\}C=\{z:Re((1-i)z)=\sqrt{2}\}$

$i)$ The number of elements in the set $A\cap B\cap C$

Answer 1

$A$ is the set of points on and above the line $y=1$ in the Argand plane.
$B$ is the set of points on the circle $(x-2{)}^2+(y-1{)}^2=9$
And
$C:Re(1-i)z=\sqrt{2}\Rightarrow Re\left(\right(1-i\left)\right(x+iy\left)\right)=\sqrt{2}\Rightarrow x+y=\sqrt{2}$

Hence, $A\cap B\cap C$ has only one point of intersection.