Question:
The complex number $z$ satisfying the condition arg $\frac {z-1} {z+1} = \frac {\pi} {4}$
The complex number $z$ satisfying the condition arg $\frac {z-1} {z+1} = \frac {\pi} {4}$
Updated On: Jul 7, 2022
- a straight line
- a circle
- a parabola
- none of these
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Let $z = x + iy$, then $\frac{z-1}{z+1}=\frac{x+iy-1}{x+iy+1}$
$=\frac{\left(x-1\right)+iy}{\left(x+1\right)+iy} \cdot \frac{\left(x+1\right)-iy}{\left(x+1\right)-iy}$
$=\frac{\left(x^{2}+y^{2}-1\right)+i\left(2y\right)}{\left(x+1\right)^{2}+y^{2}}$
Since $arg. \frac{z-1}{z+1}=\frac{\pi}{4}$
$\therefore tan \,\frac{\pi}{4}=\frac{2\,y}{x^{2}+y^{2}-1}$
$\Rightarrow 1=\frac{2y}{x^{2}+y^{2}-1}$
$\Rightarrow x^{2}+y^{2}-1=2y$
$\Rightarrow x^{2}+y^{2}-2y-1=0$
which represents a circle.
Was this answer helpful?
0
0
Top Questions on complex numbers
- Let \( z_1, z_2, z_3 \) be three complex numbers on the circle \( |z| = 1 \) with \( \arg(z_1) = -\frac{\pi}{4}, \arg(z_2) = 0 \) and \( \arg(z_3) = \frac{\pi}{4} \). If \( |z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}|^2 = \alpha + \beta \sqrt{2} \), where \( \alpha, \beta \in \mathbb{Z} \), then the value of \( \alpha^2 + \beta^2 \) is :
- JEE Main - 2025
- Mathematics
- complex numbers
- If \( Z_1 \) and \( Z_2 \) are two non-zero complex numbers, then which of the following is not true?
- KCET - 2025
- Mathematics
- complex numbers
If \( \text{Re} \left( \frac{2z + i}{z + i} \right) + \text{Re} \left( \frac{2z - i}{z - i} \right) = 2 \) is a circle of radius \( r \) and centre \( (a, b) \), then \( \frac{15ab}{r^2} \) 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
- Let the curve \(z(1 + i) +\) \(\overline{z(1 - i) = 4}\), z \(\in\)\( \mathbb{C}\), divide the region \(|z - 3| \leq 1\) into two parts of areas \( \alpha \) and \( \beta \). Then \( |\alpha - \beta| \) equals:}
- JEE Main - 2025
- Mathematics
- complex numbers
View More Questions
Notes on complex numbers
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.