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 :
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 portion of a circle centred at \((0, −\frac{1}{\sqrt3}) \) that lies in the second and third quadrants only
A portion of a circle centred at \((0, −\frac{1}{\sqrt3})\) that lies in the second only
- An empty set
A portion of a circle of radius \(\frac{2}{\sqrt3}\) that lies in the third quadrant only
The Correct Option is B
Solution and Explanation
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
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Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.