Question

If ${|z_1+z_2|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$,then
(Note : $|z_1|,|z_2|\ne 0$)

• A. $\frac{z_1}{z_2}$ is purely real
• B. $\frac{z_1}{z_2}$ is purely imaginary
• C. $z_1\overline{z_2}+z_2\overline{z_1}=0$
• D. amp $\frac{z_1}{z_2}$ may be equal to $\frac{\pi }{2}$

Answer 1

Answer: (B), (C), (D)

The correct options are
B $\frac{z_1}{z_2}$ is purely imaginary
C $z_1\overline{z_2}+z_2\overline{z_1}=0$
D amp $\frac{z_1}{z_2}$ may be equal to $\frac{\pi }{2}$

${|z_1+z_2|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$
$z_1\overline{z_2}+z_2\overline{z_1}=0$
$\Rightarrow \frac{z_1}{z_2}=-\frac{\overline{z_1}}{\overline{z_2}}$
$\Rightarrow \frac{z_1}{z_2}+\overline{\left(\frac{z_1}{z_2}\right)}=0\Rightarrow \frac{z_1}{z_2}$ is purely imaginary
so amp $\left(\frac{z_1}{z_2}\right)$ is may be $\frac{\pi }{2}$ or $-\frac{\pi }{2}$