Question

If $z_1,z_2,z_3$ are the solutions of $z^2+\overline{z}=z$, then $z_1+z_2+z_3$ is equal to (where $z$ is a complex number on the argand plane and $i=\sqrt{-1})$-

• A. $2+2i$
• B. $2-2i$
• C. $0$
• D. $2$

Answer 1

The correct option is D $2$

$z^2+\overline{z}=z......\left(i\right)$

Taking conjugate

${\overline{z}}^2+z=\overline{z}......\left(ii\right)$

adding $\left(i\right)$ and $\left(ii\right)$

$z^2+{\overline{z}}^2=0$

$\Rightarrow$ $\overline{z}=izor\overline{z}=-iz$

Case-I: $\overline{z}=iz$

$\Rightarrow$ from $\left(i\right)$

$z^2+iz=z$

$\Rightarrow$ $z=0,1-i$

Case-II: $\overline{z}=-iz$

$\Rightarrow$ $z^2-iz=z$

$\Rightarrow$ $z=0,1+i$

$\Rightarrow$ $z_1+z_2+z_3=2$