Complex Numbers and Quadratic Equations: JEE Main 2023 Mathematics question

For two non-zero complex numbers $z_{1}$ and $z_{2}$, if $\operatorname{Re}\left(z_{1} z_{2}\right)=0$ and $\operatorname{Re}\left(z_{1}+z_{2}\right)=0$, then which of the following are possible? A. $\operatorname{Im}\left(z_{1}\right)>0$ and $\operatorname{Im}\left(z_{2}\right) > 0$ B. $\operatorname{Im}\left(z_{1}\right) < 0$ and $\operatorname{Im}\left(z_{2}\right) > 0$ C. $\operatorname{Im}\left(z_{1}\right) > 0$ and $\operatorname{Im}\left(z_{2}\right) < 0$ D. $\operatorname{Im}\left(z_{1}\right) < 0$ and $\operatorname{Im}\left(z_{2}\right) < 0$ Choose the correct answer from the options given below :