${\left|z_1+z_2\right|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$ where $z_1$, $z_2$ are two complex numbers then

$z_1=x+iy{z}_2=a+ib$

${\left|z_1+z_2\right|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$

$\Rightarrow {\left(x+a\right)}^2+{\left(y+b\right)}^2=x^2+y^2+a^2+b^2$

$\Rightarrow ax+by=0$

$\frac{z_1}{z_2}=\frac{ax+by}{a^2+b^2}+\left(\frac{ay-bx}{a^2+b^2}\right)i$

$=\left(\frac{ay-bx}{a^2+b^2}\right)i$

Purely imaginary