Question

The locus of the complex number $z$ in an argand plane satisfying the equation $Arg(z+i)-Arg(z-i)=\frac{\pi }{2}$ is

• A. Boundary of a circle
• B. Interior of a circle
• C. Exterior of a circle
• D. None of these

Answer 1

The correct option is A Boundary of a circle

We have, $Arg(z+i)-Arg(z-i)=\frac{\pi }{2}$

$\Rightarrow Arg\left(\frac{z+i}{z-i}\right)=\frac{\pi }{2}$

$\therefore Re\left(\frac{z+i}{z-i}\right)=0$

$\Rightarrow \frac{\left(\frac{z+i}{z-i}\right)+\overline{\left(\frac{z+i}{z-i}\right)}}{2}=0$

$\Rightarrow \left(\frac{z+i}{z-i}\right)+\left(\frac{\overline{z}+\overline{i}}{\overline{z}-\overline{i}}\right)=0$

$\Rightarrow (z+i)(\overline{z}+i)+(z-i)(\overline{z}-i)=0$

$\Rightarrow z\overline{z}+i(z+\overline{z})-1+z\overline{z}-i(z+\overline{z})-1=0$

$\Rightarrow 2z\overline{z}=2\Rightarrow z\overline{z}=1\Rightarrow {\left|z\right|}^2=1\Rightarrow \left|z\right|=1$

The equation represents a circle centered at origin and radius $1$ unit