For two non-zero complex numbers z_1 and z_2, | vedclass

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(B) Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$, where $x_1, x_2, y_1, y_2 \in \mathbb{R}$.Given $\operatorname{Re}(z_1 + z_2) = x_1 + x_2 = 0$, w

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$B$ and $C$

Vedclass · Answer

(B) Let $z_1 = x_1 + i y_1$ and $z_2 = x_2 + i y_2$, where $x_1, x_2, y_1, y_2 \in \mathbb{R}$.Given $\operatorname{Re}(z_1 + z_2) = x_1 + x_2 = 0$, which implies $x_2 = -x_1$.Given $\operatorname{Re}(z_1 z_2) = x_1 x_2 - y_1 y_2 = 0$.Substituting $x_2 = -x_1$ into the second equation, we get $x_1(-x_1) - y_1 y_2 = 0$, which simplifies to $-x_1^2 - y_1 y_2 = 0$, or $y_1 y_2 = -x_1^2$.Since $z_1, z_2$ are non-zero, if $x_1 = 0$, then $x_2 = 0$. Consequently, $y_1 y_2 = 0$. Since $z_1, z_2 
eq 0$, this would imply $y_1 
eq 0$ and $y_2 
eq 0$, which contradicts $y_1 y_2 = 0$. Thus, $x_1 
eq 0$, so $x_1^2 > 0$.Therefore, $y_1 y_2 = -x_1^2 This indicates that $y_1$ and $y_2$ must have opposite signs.Thus, $\operatorname{Im}(z_1)$ and $\operatorname{Im}(z_2)$ have opposite signs, which corresponds to cases $(B)$ and $(C)$.

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