Question

If $arg\left(\frac{z-1}{z+1}\right)=\frac{\pi }{3}$ then interpret the locus.

Answer 1

$arg\left(\frac{z-1}{z+1}\right)=\frac{\pi }{3}$
$\Rightarrow$ $arg\left(\frac{1-z}{-1-z}\right)=\frac{\pi }{z}$
Here $arg\left(\frac{1-z}{-1-z}\right)$ represents the angle between lines joining $-1$ and $z,$ and $1$ and $z.$ As this angle is constant, the locus of $z$ will be a larger segment of circle. (angle in a segment is constant).