Question

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

Answer 1

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

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

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

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

$\Rightarrow x^2=x^2-4x+4+y^2$

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

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

Similarly,$\omega =1+s^2+2si$

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

$\Rightarrow arg(z-\omega )=\frac{\pi }{3}$

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

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

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

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