Question

If $z-1,z_2$, and $z_3$ are complex numbers such that $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=\left|\right(1/z_1)+(1/z_2)+(1/z_2\left)\right|=1$, then $|z_1+z_2+z_3|$ is

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

Answer 1

The correct option is A Equals to $1$

$|z_1|=|\overline{z_1}|=1$

$z_1\overline{z_1}=1$

Similarly, $z_2\overline{z_2}=1$ and $z_3\overline{z_3}=1$

$|\left(\frac{1}{z_1}\right)+\left(\frac{1}{z_2}\right)+\left(\frac{1}{z_3}\right)|=1$

$\Rightarrow$ $|\left(\frac{z_1\overline{z_1}}{z_1}\right)+\left(\frac{z_2\overline{z_2}}{z_2}\right)+\left(\frac{z_3\overline{z_3}}{z_3}\right)|=1$

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

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

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