Paragraph: Let $A, B, C$ be three sets of complex numbers as defined below. $A=\{z ; \operatorname{lm} z…
Paragraph:
Let $A, B, C$ be three sets of complex numbers as defined below.
$A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ; C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}$.Question:
The number of elements in the set $A \cap B \cap C$ is
0
1
2
$\infty$
Solution
Let $z=x+i y$
Set $A$ corresponds to the region $y \geq 1$.
Set $B$ consists of points lying on the circle, centred at $(2,1)$ and radius 3 , i.e., $x^2+y^2-4 x-2 y=4$
$
\text { Set } C \text { consists of points lying on the } x+y=\sqrt{2}
$
Clearly, there is only one point of intersection of the line
$
x+y=\sqrt{2} \text { and circle } x^2+y^2-4 x-2 y=4 .
$
