Mathematics Question from JEE Main 2023 · 29 January, Shift 1

For two non-zero complex numbers $z_1$ and $z_2$, if $\operatorname{Re}(z_1 z_2) = 0$ and $\operatorname{Re}(z_1 + z_2) = 0$, then which of the following are possible?

(A) $\operatorname{Im}(z_1) > 0$ and $\operatorname{Im}(z_2) > 0$

(B) $\operatorname{Im}(z_1) < 0$ and $\operatorname{Im}(z_2) > 0$

(C) $\operatorname{Im}(z_1) > 0$ and $\operatorname{Im}(z_2) < 0$

(D) $\operatorname{Im}(z_1) < 0$ and $\operatorname{Im}(z_2) < 0$

Choose the correct answer from the options given below: