[UPLIFTJEE]

For two non-zero complex numbers $\mathrm{z}_{1}$ and $\mathrm{z}_{2}$, if $\operatorname{Re}\left(\mathrm{z}_{1} \mathrm{z}_{2}\right)=0$ and $\operatorname{Re}\left(\mathrm{z}_{1}+\mathrm{z}_{2}\right)=0$, then which of the following are possible?
(A) $\operatorname{lm}\left(\mathrm{z}_{1}\right)>0$ and $\operatorname{lm}\left(\mathrm{z}_{2}\right)>0$
(B) $\operatorname{lm}\left(\mathrm{z}_{1}\right)<0$ and $\operatorname{lm}\left(\mathrm{z}_{2}\right)>0$
(C) $\operatorname{lm}\left(\mathrm{z}_{1}\right)>0$ and $\operatorname{lm}\left(\mathrm{z}_{2}\right)<0$
(D) $\operatorname{lm}\left(\mathrm{z}_{1}\right)<0$ and $\operatorname{lm}\left(\mathrm{z}_{2}\right)<0$

Choose the correct answer from the options given below : [JEE (Main) 29 Jan 2023 S I]
(A) B and D
(B) A and C
(C) A and B
(D) B and C

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[UPLIFTJEE]

Solution

$\mathrm{z}_{1}=\mathrm{x}_{1}+\mathrm{iy}_{1}, \mathrm{z}_{2}=\mathrm{x}_{2}+\mathrm{iy}_{2} \Rightarrow \operatorname{Re}\left(\mathrm{z}_{1}+\mathrm{z}_{2}\right)=\mathrm{x}_{1}+\mathrm{x}_{2}=0$
$\mathrm{x}_{1}$ and $\mathrm{x}_{2}$ are of opposite sign, $\mathrm{y}_{1}$ and $\mathrm{y}_{2}$ are of opposite sign

[UPLIFTJEE]

Answer

D

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