Let arg(z) represent the principal argument of the complex number z. Then, |z| = 3 and arg(z – 1) – arg(z + 1) = π/4 intersect
- exactly at one point
- exactly at two points
- nowhere
- at infinitely many points
The Correct Option is A
Solution and Explanation
The correct answer is (C) : nowhere
|z| = 3
arg(z-1) - arg(z+1) = π/4
\(∠AKL = ∠ACB = \frac{π}{4}\)
⇒ LK = AL = α = 1
K(0,1)
Radius = \(\sqrt2\)
PL = PK + KL = \(\sqrt2 + 1\)
P(0,1 + \(\sqrt2\))
Therefore , Number of intersection = 0
Top Questions on Complex numbers
- If \( \alpha + i\beta \) and \( \gamma + i\delta \) are the roots of the equation \( x^2 - (3-2i)x - (2i-2) = 0 \), \( i = \sqrt{-1} \), then \( \alpha\gamma + \beta\delta \) is equal to:
- JEE Main - 2025
- Mathematics
- Complex numbers
- Let integers \( a, b \in [-3, 3] \) be such that \( a + b \neq 0 \). \(\text{Then the number of all possible ordered pairs}\) \( (a, b) \), \(\text{for which}\) \[ \left| \frac{z - a}{z + b} \right| = 1 \quad \text{and} \quad \left| \begin{matrix} z + 1 & \omega & \omega^2 \\ \omega^2 & 1 & z + \omega \\ \omega^2 & 1 & z + \omega \end{matrix} \right| = 1, \] \(\text{is equal to:}\)
- JEE Main - 2025
- Mathematics
- Complex numbers
- If \( i = \sqrt{-1} \), then \[ \sum_{n=2}^{30} i^n + \sum_{n=30}^{65} i^{n+3} = \]
- AP EAPCET - 2025
- Mathematics
- Complex numbers
Let \( z \) satisfy \( |z| = 1, \ z = 1 - \overline{z} \text{ and } \operatorname{Im}(z)>0 \)
Then consider:
Statement-I: \( z \) is a real number
Statement-II: Principal argument of \( z \) is \( \dfrac{\pi}{3} \)
Then:
- AP EAPCET - 2025
- Mathematics
- Complex numbers
If \( z \) and \( \omega \) are two non-zero complex numbers such that \( |z\omega| = 1 \) and
\[ \arg(z) - \arg(\omega) = \frac{\pi}{2}, \]
Then the value of \( \overline{z\omega} \) is:
- AP EAPCET - 2025
- Mathematics
- Complex numbers
Questions Asked in JEE Main exam
- CrCl\(_3\).xNH\(_3\) can exist as a complex. 0.1 molal aqueous solution of this complex shows a depression in freezing point of 0.558°C. Assuming 100\% ionization of this complex and coordination number of Cr is 6, the complex will be:
- JEE Main - 2025
- Colligative Properties
- Suppose that the number of terms in an A.P. is \( 2k, k \in \mathbb{N} \). If the sum of all odd terms of the A.P. is 40, the sum of all even terms is 55, and the last term of the A.P. exceeds the first term by 27, then \( k \) is equal to:
- JEE Main - 2025
- Arithmetic Progression
- Let \( [x] \) denote the greatest integer function, and let \( m \) and \( n \) respectively be the numbers of the points, where the function \( f(x) = [x] + |x - 2| \), \( -2<x<3 \), is not continuous and not differentiable. Then \( m + n \) is equal to:
- JEE Main - 2025
- Differential Equations
- Let \( f(x) = \frac{x - 5}{x^2 - 3x + 2} \). If the range of \( f(x) \) is \( (-\infty, \alpha) \cup (\beta, \infty) \), then \( \alpha^2 + \beta^2 \) equals to.
- JEE Main - 2025
- Functions
- Let \( P_n = \alpha^n + \beta^n \), \( P_{10} = 123 \), \( P_9 = 76 \), \( P_8 = 47 \), and \( P_1 = 1 \). The quadratic equation whose roots are \( \frac{1}{\alpha} \) and \( \frac{1}{\beta} \) is:
- JEE Main - 2025
- Sequences and Series
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.