Question

If $z_1,z_2,z_3$ are complex numbers such that $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|=1$ , then $|z_1+z_2+z_3|$

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

Answer 1

The correct option is A Equal to 1

$z_1,z_2,z_3$ are complex numbers such that

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

$\therefore z,\overline{z_1}=1\Rightarrow \overline{z_1}=\frac{1}{z_1}$ and So on

Also $|z_1|=|\overline{z_1}|$, i.e., $\overline{z_1}$ is the conjugate of $z$,

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

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

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

$\therefore |z_1+z_2+z_3|=1$