Question

If $z_1$ and $z_2$ are non zero solutions of equation $z^2+z=i\overline{z}$ where $i=\sqrt{-1}$ , then the value of $|z_1+z_2|$ is

Answer 1

$z^2+z=i\overline{z}$
$\Rightarrow z(z+1)=i\overline{z}\dots \left(1\right)$

Taking modulus both the sides, we get
$\Rightarrow \left|z\right||z+1|=\left|z\right|$
$\Rightarrow |z+1|=1(\because |z|\ne 0)$
$\Rightarrow (z+1)(\overline{z}+1)=1$
$\Rightarrow \overline{z}=\frac{-z}{z+1}$

Multiplying by $i$ both the sides, we get
$\Rightarrow i\overline{z}=i\times \frac{-z}{z+1}$
$\Rightarrow z(z+1)=i\left(\frac{-z}{z+1}\right)\left[\text{from (1)}\right]$
$\Rightarrow (z+1{)}^2=-i=e^{-i\pi /2}\Rightarrow z+1=\pm {\left(e^{-i\pi /2}\right)}^{1/2}=\pm e^{-i\pi /4}$
$\Rightarrow z+1=\pm \frac{1-i}{\sqrt{2}}$
$\Rightarrow z_1=\frac{1-i}{\sqrt{2}}-1$
and $z_2=-\frac{1-i}{\sqrt{2}}-1$
$\therefore z_1+z_2=-2$
$\Rightarrow |z_1+z_2|=2$