Mathematics - Complex Number Question with Solution | TestHub

Set $A$ is $\left\{z\in C:\left|\frac{z+1}{z-1}\right|<1\right\}$

$\Rightarrow \left|\frac{\overline{z}+1}{\overline{z}-1}\right|<1$

$\Rightarrow \left|\overline{z}+1\right|<\left|\overline{z}-1\right|$

$\Rightarrow {\left(x+1\right)}^2+y^2<{\left(x-1\right)}^2+y^2$

$\Rightarrow x<0$

Set $B$ is $\left\{z\in C:\arg \left(\frac{z-1}{z+1}\right)=\frac{2\pi }{3}\right\}$

$\Rightarrow \arg \left(\frac{z-1}{z+1}\right)=\frac{2\pi }{3}$

$\Rightarrow {\tan }^{-1}\left(\frac{y}{x-1}\right)-{\tan }^{-1}\left(\frac{y}{x+1}\right)=\frac{2\pi }{3}$

$\Rightarrow x^2+y^2+\frac{2y}{\sqrt{3}}-1=0$

$\Rightarrow$  Centre $\left(0,-\frac{1}{\sqrt{3}}\right)$

Hence, $A\cap B$ will represent a portion of a circle centred at $\left(0,-\frac{1}{\sqrt{3}}\right)$ that lies in the second quadrant only