Let S be the set of all complex numbers z satisfying z-2+i geq √(5) . If the complex number z 0 is such that frac 1 z 0 -1 is the maximum of the set \ 1/ z-1 : z in S \ , then the principal argument of frac 4-z 0 -overline z 0 z 0 -overline z 0 +2 i is

Mathematics · Complex Numbers

JEE Advanced 2019 — Paper 1 — Question 31

Let $S$ be the set of all complex numbers $z$ satisfying $|z-2+i| \geq \sqrt{5}$ . If the complex number $z_{0}$ is such that $\frac{1}{\left|z_{0}-1\right|}$ is the maximum of the set $\left\{\frac{1}{|z-1|}: z \in S\right\}$ , then the principal argument of $\frac{4-z_{0}-\overline{z_{0}}}{z_{0}-\overline{z_{0}}+2 i}$ is

Option A:
$\frac{3 \pi}{4}$
Option B:
$\frac{\pi}{4}$
Option C:
$-\frac{\pi}{2}$
Correct
Option D:
$\frac{\pi}{2}$

Answer: C

Step-by-step solution

Clearly location of required point $\mathrm{z}_{0}$ is at P with abscissa $<1 \&$ ordinate $>0$

Now $\arg \left[\frac{4-z_{0}-\bar{z}_{0}}{z_{0}-\bar{z}_{0}+z i}\right]=\operatorname{Arg}\left(\frac{x-2}{y+1}\right) i=\operatorname{Arg}$ ki $\& k<0$

$\Rightarrow$ Required argument $=-\pi / 2$

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2019
Paper: Paper 1
Subject: Mathematics
Chapter: Complex Numbers
Topic: Properties of Complex Numbers

Let omega neq 1 be a cube root of unity. Then the minimum of the set \ a+b omega+c omega^2 ^2: a, b, c . distinct non-zero integers \…

Answer: C

Step-by-step solution

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