Question

If $z_1,z_2,z_3$ are imaginary numbers such that $|z_1|=|z_2|=|z_3|=\left|\begin{array}{c}\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}\end{array}\right|=1$ then $|z_1+z_2+z_3|$ is

• A. Equal to $1$
• B. Less than $1$
• C. Greater than $1$
• D. Equal to $3$

Answer 1

The correct option is A Equal to $1$

using $z\overline{z}=|z{|}^2$

$z_1\overline{z_1}=|z_1{|}^2=1\Rightarrow \frac{1}{z_1}=\overline{z_1}$

$z_2\overline{z_2}=|z_2{|}^2=1\Rightarrow \frac{1}{z_2}=\overline{z_2}$

and $z\overline{z_3}=|z_3{|}^2=1\Rightarrow \frac{1}{z_3}=\overline{z_3}$

Now using, $|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|=1$

$\Rightarrow |{\overline{z}}_1+{\overline{z}}_2+{\overline{z}}_3|=1$

$\Rightarrow |\overline{z_1+z_2+z_3}|=1$

$\therefore |z_1+z_2+z_3|=1$, since $|\overline{z}|=|z|$