Question

Let $z_1$ and $z_2$ be complex numbers such that $z_1\ne z_2$ and $|z_1|$ = $|z_2|$. 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
• B. Real and positive
• C. Real and negative
• D. Purely imaginary or zero

Answer 1

The correct option is D Purely imaginary or zero

$\frac{z_1+z_2}{z_1-z_2}.\frac{\overline{z_1}-\overline{z_2}}{\overline{z_1}-\overline{z_2}}=\frac{2iIm\left(\overline{z_1}{z}_2\right)}{|z_1-z_2{|}^2}$
$\Rightarrow$ Purely imaginary, when Im$(z_1\overline{z_1}\ne 0)$ or zero