Question

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

$iii)$ Let $z$ be any point in $A\cap B\cap C$ and $w$ be any point satisfying $|w-2-i|<3$. Then $|z|-|w|+3$ lies between

Answer 1

$||z|-|w||<|z-w|$ and $|z-w|$ is the distance between $z$ and $w$
Here, $z$ is fixed. Hence distance between $z$ and $w$ would be maximum for diametrically opposite points. Therefore,
$|z-w|<6$
$\Rightarrow -6<|z|-|w|<6$
$\Rightarrow -3<|z|-|w|+3<9$