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

The Correct Option is B

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.