Let $zand\omega$ be  complex numbers. If $Re\left(z\right)=|z-2|$ ,  $Re\left(\omega \right)=|\omega -2|$ and  $arg(z-\omega )=\frac{\pi }{3},$then value of $\mathrm{Im}(z+\omega )$ is

Let   $z=x+iy,x,y\in R$ and  $i=\sqrt{-1}$

$\because \mathrm{Re}\left(z\right)=|z-2|$

$\Rightarrow x=\overline{){(x-2)}^2+y^2}$

$\Rightarrow y^2=4(x-1)$

$z=1+t^2+2ti$ parametric from and let $\omega =p+iq$

Similarly $\omega =1+s^2+2si$

$z-\omega =(t^2-s^2)+2i(t-s)$

arg $(z-\omega )=\frac{\pi }{3}$

${\tan }^{-1}\left(\frac{2}{t+s}\right)=\frac{\pi }{3}\Rightarrow \frac{2}{t+s}=\sqrt{3}$

$(t+s)=\frac{2}{\sqrt{3}}$

Now        $z+\omega =2+t^2+s^2+2i(t+s)$

$\mathrm{Im}(z+\omega )=2(t+s)=\frac{4}{\sqrt{3}}$