Question

If $z_1$ and $z_2$ are two non-zero complex numbers such that $|z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2$. then

• A. $z_1\overline{z_2}$ is purely imaginary
• B. $z_1/z_2$ is purely imaginary
• C. $z_1{\overline{z}}_2+{\overline{z}}_1{z}_2=0$
• D. $0,z_1,z_2$ are the vertices of a right triangle

Answer 1

The correct option is A $z_1\overline{z_2}$ is purely imaginary

$|z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2+2Re\left(z_1{\overline{z}}_2\right)$

$|z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2\Rightarrow Re\left(z_1{\overline{z}}_2\right)=0$

Thus, $z_1{\overline{z}}_2$ is purely imaginary. So, A is the correct option.

Since $\frac{1}{z_2}=\frac{{\overline{z}}_2}{|z_2{|}^2}$

$\frac{z_1}{z_2}=\frac{z_1{\overline{z}}_2}{|z_2{|}^2}$

Since the magnitude of a complex number is a real number $\Rightarrow$ Option B is also correct.