Question

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

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

Answer 1

The correct option is A Equal to 1

$1=|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|=|\frac{z_1\overline{z_1}}{z_1}+\frac{z_2\overline{z_2}}{z_2}+\frac{z_3\overline{z_3}}{z_3}|$
$[\because |{\overline{z_1}}^2=1=z_1\overline{z_1}]$
$=|\overline{z_1}+\overline{z_2}+\overline{z_3}|=|\overline{z_1+z_2+z_3}|=|z_1+z_2+z_3|$
$[\because |\overline{z_1}|=|z_1|]$