Question

If $z$ is a complex number such that $\arg \left(\frac{ z -1}{ z +1}\right)=\frac{\pi}{3}$, then find the value of $\sqrt{3}\left| z -\frac{1}{\sqrt{3}} i\right|$.

Answer 1

If $z=x+i y$, then
$\operatorname{Arg}( z -1)=\tan ^{-1}\left(\frac{ y }{ x -1}\right) $
$\operatorname{Arg}( z +1)=\tan ^{-1}\left(\frac{ y }{ x +1}\right) $
$\operatorname{Arg}( z -1)-\operatorname{Arg}( z +1)=\tan ^{-1}\left(\frac{ y }{ x -1}\right)-\tan ^{-1}\left(\frac{ y }{ x +1}\right) $
$\frac{\pi}{3}=\tan ^{-1}\left(\frac{2 y }{1+ x ^2+ y ^2}\right) \Rightarrow x ^2+ y ^2-\frac{2}{\sqrt{3}} y =1 \Rightarrow x ^2+\left( y -\frac{1}{\sqrt{3}}\right)^2=\frac{4}{3}$
$\Rightarrow \left| z -\frac{i}{\sqrt{3}}\right|^2=\frac{4}{3} \Rightarrow \sqrt{3}\left| z -\frac{i}{\sqrt{3}}\right|=2$