Question

The perimeter of the locus represented by $\arg \left(\frac{z+i}{z-i}\right)=\frac{\pi }{4}$ is equal to

• A. $4\pi$
• B. $2\sqrt{2}\pi$
• C. $2\pi \sqrt{3}$
• D. $\frac{2\pi }{3}$

Answer 1

The correct option is B $2\sqrt{2}\pi$

Let $\alpha$ be a complex number defined as,
$\alpha =\left(\frac{z+i}{z-i}\right)$
Assuming $z=x+iy$
$\alpha =\frac{x+i(y+1)}{x+i(y-1)}\Rightarrow \alpha =\frac{x+i(y+1)}{x+i(y-1)}\times \frac{x-i(y-1)}{x-i(y-1)}$
So the argument of the complex number will be,
$\tan \frac{\pi }{4}=\frac{Im\left(\alpha \right)}{Re\left(\alpha \right)}\Rightarrow \frac{x(y+1)-x(y-1)}{x^2+(y-1)(y+1)}=1\Rightarrow x^2-2x+y^2=1\Rightarrow (x-1{)}^2+y^2=2$
So the locus is a circle with radius $\sqrt{2}$
Perimeter $=2\pi r=2\sqrt{2}\pi$