Question

Let $A=\left\{z\in C:\left|\frac{z+1}{z-1}<1\right|\right\}$ and $B=\left\{z\in C:\arg \left(\frac{z-1}{z+1}\right)=\frac{2\pi }{3}\right\}$ Then $A\cap B$ is :

• A. a portion of a circle centred at $\left(0,-\frac{1}{\sqrt{3}}\right)$ that lies in the second and third quadrants only
• B. a portion of a circle centred at $\left(0,-\frac{1}{\sqrt{3}}\right)$ that lies in the second quadrant only
• C. an empty set
• D. a portion of a circle of radius $\frac{2}{\sqrt{3}}$ that lies in the third quadrant only

Answer 1

Answer: (B)

Set $A$
$\Rightarrow \left|\frac{z+1}{z-1}\right|<1$
$\Rightarrow |z+1|<|z-1|$
$\Rightarrow (x+1{)}^2+y^2<(x-1{)}^2+y^2$
$\Rightarrow x<0$

$\Rightarrow \arg \left(\frac{z-1}{z+1}\right)=\frac{2\pi }{3}$
$\Rightarrow {\tan }^{-1}\left(\frac{y}{x-1}\right)-{\tan }^{-1}\left(\frac{y}{x+1}\right)=\frac{2\pi }{3}$
$\Rightarrow x^2+y^2+\frac{2y}{\sqrt{3}}-1=0$
$A\cap B$
$\Rightarrow$ Centre $\left(0,-\frac{1}{\sqrt{3}}\right)$