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  4. For two non-zero complex numbers z1 and z2, if textRe(z1 z2)=0 and textRe(z1+z2)=0, then which of the following are possible? A. textIm(z1)>0 and textIm(z2)>0 B. textIm(z1)<0 and textIm(z2)>0 C. textIm(z1)>0 and textIm(z2)<0 D. textIm(z1)<0 and textIm(z2)<0 Choose the correct answer from the options given below:

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Q. For two non-zero complex numbers $z_1$ and $z_2$, if $\text{Re}\left(z_1 z_2\right)=0$ and $\text{Re}\left(z_1+z_2\right)=0$, then which of the following are possible?
A. $\text{Im}\left(z_1\right)>0$ and $\text{Im}\left(z_2\right)>0$
B. $\text{Im}\left(z_1\right)<0$ and $\text{Im}\left(z_2\right)>0$
C. $\text{Im}\left(z_1\right)>0$ and $\text{Im}\left(z_2\right)<0$
D. $\text{Im}\left(z_1\right)<0$ and $\text{Im}\left(z_2\right)<0$
Choose the correct answer from the options given below:

JEE MainJEE Main 2023Complex Numbers and Quadratic Equations

Solution:

$z_1=x_1+i y_1$
$z _2= x _2+ i y _2$
$\operatorname{Re}\left( z _1 z _2\right)= x _1 x _2- y _1 y _2=0$
$\operatorname{Re}\left( z _1+ z _2\right)= x _1+ x _2=0$
$x _1 \& x _2$ are of opposite sign
$y_1 \& y_2$ are of opposite sign