Question

lf $z_1,z_2,...,z_n$ lie on the circle $|z|=2$, then the value of $|z_1+z_2+\dots +z_n|-4|\frac{1}{z_1}+\frac{1}{z_2}+\dots +\frac{1}{z_n}|$ is

• A. $0$
• B. $n$
• C. $-n$
• D. $1$

Answer 1

The correct option is A $0$

|1/$z_1$ + 1/$z_2$ + 1/$z_3$ + ... + 1/$z_n$ |

= |$\overline{z_1}/{|z_1|}^2$ + $\overline{z_2}/{|z_2|}^2$ + ... + $\overline{z_n}/{|z_n|}^2$ | (Multiplying and dividing by conjugate)

=$\frac{1}{4}$ $|\overline{z_1}+\overline{z_2}+...+\overline{z_n}|$ (since |z| = 2 and $z_1$ , $z_2$ , ... lies on this circle)

Putting it into given equation gives result equals $0$.

Hence, Option A is correct