Question

Let $A,B,C$ be three sets of complex numbers as defined below.
$A=\{z:\text{Imz}\ge 1\}$
$B=\{z:|z-2-i|=3\}$
$C=\{z:\text{Re}((1-i)z)=\sqrt{2}\}$
Let $z$ be any point in $A\cap B\cap C$ and let $w$ be any point satisfying $|w-2-i|<3$. Then, $|z|-|w|+3$ lies between

• A. -6 and 3
• B. -3 and 6
• C. -6 and 6
• D. -3 and 9

Answer 1

Answer: (D)

$\Vert z|-|w\Vert <|z-w|$
and $|z-w|=$ Distance between $z$ and $w$
$z$ is fixed.
Hence distance between $z$ and $w$ would be maximum for diametrically opposite points.
$\Rightarrow |z-w|<6$
$\Rightarrow -6<|z|-|w|<6$
$\Rightarrow -3<|z|-|w|+3<9$