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: 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
  1. $-6$ and 3
  2. $-3$ and 6
  3. $-6$ and 6
  4. $-3$ and 9

Solution

$ \begin{array}{ll} & |w-(2+i)| < 3 \\ \Rightarrow & \quad|| w|-| 2+i|| < 3 \\ \Rightarrow & -3+\sqrt{5} < |w| < 3+\sqrt{5} \\ \Rightarrow & -3-\sqrt{5} < -|w| < 3-\sqrt{5} \end{array} $ Also, $|z-(2+i)|=3$ $ \begin{aligned} \Rightarrow \quad-3+\sqrt{5} & \leq|z| \leq 3+\sqrt{5} \\ -3 & < |z|-|w|+3 < 9 \end{aligned} $

Asked in: JEE Advanced 2008 (Paper 1)

Practice more Complex Number questions on Aicharya