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:
