Complex Numbers 2 Question 27
28. 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
Passage II
Let $S=S _1 \cap S _2 \cap S _3$, where
$S _1={z \in C:|z|<4}, S _2=z \in C: \operatorname{lm} \frac{z-1+\sqrt{3} i}{1-\sqrt{3} i}>0$
and $S _3:{z \in C: \operatorname{Re} z>0}$
(2008)
Show Answer
Answer:
Correct Answer: 28. $x=2 n \pi+2 \alpha, \alpha=\tan ^{-1} k$, where $k \in(1,2)$ or $x=2 n \pi$
Solution:
- Since,
$$ |w-(2+i)|<3 \Rightarrow|w|-|2+i|<3 $$
$$ \begin{array}{ll} \Rightarrow & -3+\sqrt{5}<|w|<3+\sqrt{5} \\ \Rightarrow & -3-\sqrt{5}<-|w|<3-\sqrt{5} \end{array} $$
Also, $\quad|z-(2+i)|=3$
$$ \begin{array}{ll} \Rightarrow & -3+\sqrt{5} \leq|z| \leq 3+\sqrt{5} \\ \therefore & -3<|z|-|w|+3<9 \end{array} $$
