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Q.
Let $S=\left\{z \in C : z^2+\bar{z}=0\right\}$.
Then $\displaystyle\sum_{z=S}(\operatorname{Re}(z)+\operatorname{Im}(z))$ is equal to _______
JEE MainJEE Main 2022Complex Numbers and Quadratic Equations
Solution:
$S=\left\{z \in C: z^2+\bar{z}=0\right\}$
Let $z = x + iy$
$ z ^2= x ^2- y ^2+2 ixy$
$ \bar{z}=x-i y$
$z^2+\bar{z}=x^2-y^2+x+i(2 x y-y)=0$
$ \Rightarrow x^2+x-y^2=0 \& 2 x y-y=0 $
$y=0 \text { or } x=\frac{1}{2}$
If $y =0 ; x =0,-1$
If $x=\frac{1}{2} ; y=\frac{\sqrt{3}}{2}, \frac{-\sqrt{3}}{2}$
$\displaystyle\sum_{z \in S}\left(\operatorname{Re}(z)+\operatorname{Im}(z)=\left(0-1+\frac{1}{2}+\frac{1}{2}\right)+0+0+\frac{\sqrt{3}}{2}-\frac{\sqrt{3}}{2}\right)$