Question

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

• A. A portion of a circle centred at $(0, −\frac{1}{\sqrt3})$ that lies in the second and third quadrants only
• B. A portion of a circle centred at $(0, −\frac{1}{\sqrt3})$ that lies in the second only
• C. An empty set
• D. A portion of a circle of radius $\frac{2}{\sqrt3}$ that lies in the third quadrant only

Answer 1

Answer: (B)

A portion of a circle centred at $(0, −\frac{1}{\sqrt3})$ that lies in the second only

Answer 2

The correct answer is (B) : A portion of a circle centred at $(0, −\frac{1}{\sqrt3})$ that lies in the second only
$|\frac{z+1}{z−1}|<1⇒|z+1|<|z−1|⇒Re(z)<0$
and $arg(\frac{z−1}{z+1})=\frac{2π}{3}$
is a part of circle as shown

Fig.