Question

Let $z$ be a complex number lying in first or fourth quadrant of Argand plane satisfying $|z-1|=1.$ If $\arg (z-1)=k\arg \left(z\right),$ then the value of $k$ is

Answer 1

From polar form,
$z-1=\cos \theta +i\sin \theta (\because |z-1|=1)$
$\Rightarrow z=1+\cos \theta +i\sin \theta$
$=2{\cos }^2\frac{\theta }{2}+2i\sin \frac{\theta }{2}\cos \frac{\theta }{2}$
$=2\cos \frac{\theta }{2}(\cos \frac{\theta }{2}+i\sin \frac{\theta }{2})$
$\therefore \arg \left(z\right)=\frac{\theta }{2}=\frac{1}{2}\arg (z-1)$
Hence, $\arg (z-1)=2\arg \left(z\right)$
$\therefore k=2$