Question

Let $A,B,C$ be three sets of complex number defined as $A=\{z:Im(z)\ge 1\},B=\{z:|z-2-i|=3\},C=\{z:Re((1-i)z)=\sqrt{2}\}.$
If $z$ be any point in $A\cap B\cap C,$ then $|z+1-i{|}^2+|z-5-i{|}^2$ lies between

• A. $25$ and $29$
• B. $30$ and $34$
• C. $35$ and $39$
• D. $40$ and $44$

Answer 1

The correct option is C $35$ and $39$


Set $A$ is the region $y\ge 1$.
Set $B$ is a circle with at $(2,1)$ and radius $3\text{units}$.
Set $C$ is the straight line $x+y=\sqrt{2}$.
$A\cap B\cap C$ is a point $z$ shown in the figure.


So, $|z+1-i{|}^2+|z-5-i{|}^2=(\text{diameter}{)}^2=6^2=36$