Question

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

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