Let $z_1$ and $z_2$ be two complex numbers such that $z_1≠z_2$ and $|z_1|\neq|z_2|$. If $z_1$ has positive real part and $z_2$ has negative imaginary part, then $\frac{z_1+z_2}{z_1−z_2}$ may be ___________
a) zero or purely imaginary
b) real and +ve
c) real and -ve
My reference gives the solution "zero or purely imaginary". If it was $|z_1|=|z_2|$ I can easily find the solution from geometry as the complex numbers $z_1$ and $z_2$ form a rhombus and $z_1+z_2$ and $z_1-z_2$ be its diagonals and they intersect orthoganally, thus $\arg\Big(\frac{z_1+z_2}{z_1-z_2}\Big)=\arg(z_1+z_2)-\arg(z_1-z_2)=\pm\frac{\pi}{2}$, thus purely imaginary. But, in this case how do I find the solution ?