Question

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

• A. $0$
• B. $n$
• C. $-n$
• D. $1$

Answer 1

The correct option is A $0$

As $z_1,z_2,\cdots z_n$ lie on the circle $|z|=2,$
$|z_i|=2\Rightarrow |z_i{|}^2=4$
$\Rightarrow z_i{\overline{z}}_i=4$ for $i=1,2,3,\cdots ,n$
Thus $\frac{1}{z_i}=\frac{1}{4}{\overline{z}}_i$ for $i=1,2,\cdots ,n.$
So, $|z_1+z_2+\cdots +z_n|-4|\frac{1}{z_1}+\frac{1}{z_2}+\cdots +\frac{1}{z_n}|$
$\Rightarrow |z_1+z_2+\cdots +z_n|-4|\frac{1}{4}{\overline{z}}_1+\frac{1}{4}{\overline{z}}_1+\cdots +\frac{1}{4}{\overline{z}}_n|$
$=|z_1+z_2+\cdots +z_n|-|{\overline{z}}_1+{\overline{z}}_1+\cdots +{\overline{z}}_n|$
$=|z_1+z_2+\cdots +z_n|-\left|\overline{z_1+z_1+\cdots +z_n}\right|$
$=0$ $[\because |z|=|\overline{z}|]$