Question

If $|z_1+z_2{|}^2$ = $|z_1-z_2{|}^2$ where $z_1$ and $z_2$ are non-zero complex numbers, then :

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

$|z_1+z_2{|}^2$ =$|z_1{|}^2+|z_2{|}^2$ + $2cos(\alpha -\beta )$ and $|z_1-z_2{|}^2$ = $|z_1{|}^2+|z_2{|}^2-2cos(\alpha -\beta )$

Where $\alpha$ = Arg$z_1$ and $\beta$ = Arg$z_2$

If $|z_1+z_2{|}^2$ = $|z_1-z_2{|}^2$ then $cos(\alpha -\beta )$ = 0 $\Rightarrow$ $\alpha$ - $\beta$ = $\frac{\pi }{2}$

Arg$z_1$ - Arg$z_2$ = $\frac{\pi }{2}$ $\Rightarrow$ Arg $\frac{z_1}{z_2}$ = $\frac{\pi }{2}$ so Re$\left(\frac{z_1}{z_2}\right)$ = 0