IIT-JEE 1986Question 1
Let $z_1$ and $z_2$ be complex numbers such that $z_1\neq z_2$ and $|z_1|=|z_2|$. If $z_1$ has positive real part and $z_2$ has negative imaginary part, then $\dfrac{z_1+z_2}{z_1-z_2}$ may be
Complex Numbers
Solution
Put $z_1=re^{i\alpha}$, $z_2=re^{i\beta}$: $\frac{z_1+z_2}{z_1-z_2}=-i\cot\frac{\alpha-\beta}{2}$, purely imaginary (zero when $z_2=-z_1$, allowed e.g. $z_1=e^{i\pi/3}$, $z_2=e^{-2i\pi/3}$). A nonzero real value is impossible since the real part is always $0$.