Perimeter of the locus represented by $\arg \left(\frac{z+i}{\left(z-i\right)}\right)=\frac{\pi }{4}$ , (where $i=\sqrt{-1}$ ) is equal to:

 $\because$                       $\angle QPR=\pi /4$ 

$\therefore \angle QCR=\pi /2$ , where $C$  center of circle.

Now in $\Delta QCR$ .

              $r^2+r^2={\left(QR\right)}^2={\left|2i\right|}^2=4$

$\therefore r=\sqrt{2}$

$\therefore$    Required perimeter $=2\pi r-\frac{1}{4}\left(2\pi r\right)$

           $=\frac{3}{2}\pi r=\frac{3}{2}\pi \left(\sqrt{2}\right)=\frac{3\pi }{\sqrt{2}}$ .