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 $|z_1+z_2+z_3|$ 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

Let $z_1=cis{\theta }_1$, $z_2=cis{\theta }_2$ and $z_3=cis{\theta }_3$

We have,

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

$|cis(-{\theta }_1)+cis(-{\theta }_2)+cis(-{\theta }_3)|=1$

Hence,

$|(\cos {\theta }_1+\cos {\theta }_2+\cos {\theta }_3)-i(\sin {\theta }_1+\sin {\theta }_2+\sin {\theta }_3)|=1$

Hence,

$|(\cos {\theta }_1+\cos {\theta }_2+\cos {\theta }_3)+i(\sin {\theta }_1+\sin {\theta }_2+\sin {\theta }_3)|=1$

Hence,

$|z_1+z_2+z_3|=1$