Question

Consider the complex number $z_1$ and $z_2$ satisfying the relation $|z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2$, then the possible difference between the argument of $z_1$ and $z_2$ is,

• A. $0$
• B. $\pi$
• C. $-\frac{\pi }{2}$
• D. none of these

Answer 1

The correct option is C $-\frac{\pi }{2}$

Given that,

$|z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2$

$\Rightarrow |z_1{|}^2+|z_2{|}^2+z_1{\overline{z}}_2+{\overline{z}}_1{z}_2=|z_1{|}^2+|z_2{|}^2$

$\Rightarrow z_1{\overline{z}}_2+{\overline{z}}_1{z}_2=0$

$\Rightarrow \frac{z_1}{z_2}+\frac{{\overline{z}}_1}{{\overline{z}}_2}=0$

(divided by $z_2{\overline{z}}_2)$

$\Rightarrow \frac{z_1}{z_2}+\left(\overline{\frac{z_1}{z_2}}\right)=0$

$z_1/z_2$ is purely imaginary.

Hence, $arg\left(\frac{z_1}{z_2}\right)=\pm \frac{\pi }{2}$