${|z_1+z_2|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$ where z₁, z₂ are two complex numbers then

$$\begin{gathered} z_1=x+iy{z}_2=a+ib \\ {|z_1+z_2|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2 \end{gathered}$$
$$\begin{gathered} \Rightarrow {(x+a)}^2+{(y+b)}^2=x^2+y^2+a^2+b^2 \\ \Rightarrow ax+by=0 \\ \frac{z_1}{z_2}=\frac{ax+by}{a^2+b^2}+\left(\frac{ay-bx}{a^2+b^2}\right)i \\ =\left(\frac{{\displaystyle ay-bx}}{{\displaystyle a^2+b^2}}\right)i \end{gathered}$$
Purely imaginary