Problem 142 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(z_{1}\) and \(z_{2}\) both satisfy the relation \(z+\bar{z}=2|z-1|\) and arg \(\left(z_{1}-z_{2}\right)=\frac{\pi}{4}\), then the imaginary part of \(\left(z_{1}+z_{2}\right)\) is (A) 0 (B) 1 (C) 2 (D) None of these

Step-by-Step Solution

Answer

The imaginary part of \(z_1 + z_2\) is 0. (A)

Step 1: Understand the equation

We have the equation \( z + \bar{z} = 2|z - 1| \). Recognize that \( \bar{z} \) is the conjugate of \( z \), so \( z + \bar{z} = 2\text{Re}(z) \). This means \( 2\text{Re}(z) = 2|z - 1| \); simplify to \( \text{Re}(z) = |z - 1| \).

Step 2: Express complex number as x+iy

Let \( z = x + iy \) where \( x, y \) are real numbers. Then \( \overline{z} = x - iy \). The expression becomes \( x = |z - 1| \), where \(|z - 1| = \sqrt{(x-1)^2 + y^2}\).

Step 3: Solve for x and y

From \( x = \sqrt{(x-1)^2 + y^2} \), square both sides to get \( x^2 = (x-1)^2 + y^2 \). Simplify to get \( x^2 = x^2 - 2x + 1 + y^2 \). This reduces to \( 2x = 1 + y^2 \), which further gives \( y^2 = 2x - 1 \).

Step 4: Consider the argument condition

Given \( \text{arg}(z_1 - z_2) = \frac{\pi}{4} \), it means \( z_1 - z_2 \) forms a \( 45^{\circ} \) angle with the real axis. This implies \( \frac{\text{Im}(z_1 - z_2)}{\text{Re}(z_1 - z_2)} = 1 \), i.e., \( \text{Im}(z_1 - z_2) = \text{Re}(z_1 - z_2) \).

Step 5: Analyze z_1 and z_2 satisfying the condition

For both \( z_1 = x_1 + iy_1 \) and \( z_2 = x_2 + iy_2 \) satisfying \( x = \sqrt{(x-1)^2 + y^2} \), we use the derived relation \( y^2 = 2x - 1 \).

Step 6: Determine the imaginary part of (z1 + z2)

When we use the equality \( \text{Im}(z_1 - z_2) = \text{Re}(z_1 - z_2) \), let the real parts be \( x_1 = x_2 = x \), and hence \( y_1^2 = y_2^2 \) because they follow \( y_i^2 = 2x - 1 \) for both, thus \( y_1 + y_2 = |y_1| + |y_2| = 1 + 1 = 2 \). Therefore, \( \text{Im}(z_1 + z_2) = 0 \).

Step 7: Interpret logical outcomes

Setting \( y_1 = 1 \) and \( y_2 = -1 \), for example, satisfies the argument condition, but their sum is 0. Thus, the imaginary component of their sum is 0.

Key Concepts

Argand Diagram

In studying complex numbers, the Argand Diagram is a visual representation used frequently. It plots complex numbers on a plane, where the horizontal axis represents the real part and the vertical axis corresponds to the imaginary part. This diagram allows us to visualize complex numbers as vectors.
For example, if a complex number is represented as \( z = x + iy \), then it is depicted on the Argand Diagram as a point \( (x, y) \).
Placing complex numbers on this diagram helps to easily interpret operations on complex numbers, such as addition or multiplication, by simple geometric transformations. Understanding the Argand Diagram is crucial for dealing with equations involving complex numbers, like the ones using the expression \( z + \bar{z} \). It allows us to see the sum of a complex number and its conjugate, aiding in realizing that this is twice the real part.

Complex Conjugate

The concept of a complex conjugate involves changing the sign of the imaginary part of a complex number. For a complex number \( z = x + iy \), its conjugate is \( \bar{z} = x - iy \).
The usefulness of complex conjugates shows up in many calculations:

• They help simplify expressions involving complex numbers.
• Multiplying a complex number by its conjugate yields a real number, specifically \( |z|^2 \), the square of the magnitude.
• In our original exercise, the fact that \( z + \bar{z} = 2 \text{Re}(z) \) is directly related to the behavior of conjugates.

Understanding conjugates is essential for decomposing complex numbers into real and imaginary components, and divorcing the effects of phase (argument) from the magnitude (modulus).

Argument of a Complex Number

The argument of a complex number, often denoted as \( \text{arg}(z) \), represents the angle that the line representing the complex number forms with the positive real axis on the Argand Diagram.
Expressed in radians or degrees, it provides a sense for the direction of the vector rather than its length. For example:

• In the problem provided, the condition that the argument of \( z_1 - z_2 \) is \( \frac{\pi}{4} \) or 45 degrees, indicates that the difference between these numbers is oriented such that it forms a 45-degree angle with the real axis.
• This implies a specific relationship between the real and imaginary parts, where \( \text{Im}(z_1-z_2) = \text{Re}(z_1-z_2) \).

Knowing how to calculate and interpret arguments is key in analyzing and comparing complex numbers geometrically.

Real and Imaginary Parts

Every complex number \( z = x + iy \) is comprised of two key components: the real part \( x \) and the imaginary part \( y \). In this format:

• The real part \( x \) lies along the horizontal axis on the Argand Diagram.
• The imaginary part \( y \) projects along the vertical axis.

In the equation from our problem \( z + \bar{z} = 2|z - 1| \), isolating the real part leads to \( 2\text{Re}(z) = 2|z - 1| \), simplifying to \( \text{Re}(z) = |z - 1| \).
Subsequently, when solving for \( x \) and \( y \), the relationship \( y^2 = 2x - 1 \) is derived. This relationship links the real and imaginary components, shedding light on their interdependence and revealing conditions they must satisfy, such as how their combination in \( z_1 + z_2 \) results in an imaginary part of 0.