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 |$z_1$+$z_2$+$z_3$|

• 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=$|\frac{1}{z_1}+\frac{1}{z_2}+\frac{1}{z_3}|$=$|\frac{z_1\ast {\overline{z}}_1}{z_1}+\frac{z_2\ast {\overline{z}}_2}{z_2}+\frac{z_3\ast {\overline{z}}_3}{z_3}|$

(hence, ${|z_1|}^2$=1=$z_1$$\overline{z_1}$, etc)

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

(hence,|$\overline{z_1}$|=|$z_1$|)