Problem 212 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(z\) is a complex number of unit modulus and argument \(\theta\), then \(\left(\frac{1+z}{1+\bar{z}}\right)\) equals \(\quad\) [2013] (A) \(\frac{\pi}{2}-\theta\) (B) \(\theta\) (C) \(\pi-\theta\) (D) \(-\theta\)

Step-by-Step Solution

Answer

The expression simplifies to \(e^{i\theta}\), so the answer is (B) \(\theta\).

Step 1: Understanding the Given Information

We know that the complex number \(z\) has unit modulus and argument \(\theta\). This means \(|z| = 1\) and \(z = e^{i\theta}\).

Step 2: Finding the Conjugate of z

The conjugate of a complex number \(z = e^{i\theta}\) is denoted as \(\bar{z}\) and is equal to \(e^{-i\theta}\).

Step 3: Expressing Given Expression

We need to evaluate \(\left(\frac{1+z}{1+\bar{z}}\right)\), where \(z = e^{i\theta}\) and \(\bar{z} = e^{-i\theta}\).

Step 4: Substitute z and \(\bar{z}\) in the Expression

Substitute \(z = e^{i\theta}\) and \(\bar{z} = e^{-i\theta}\) into the expression: \[ \frac{1 + e^{i\theta}}{1 + e^{-i\theta}}. \]

Step 5: Simplifying the Expression

Multiply the numerator and the denominator by \(e^{i\theta}\). This step will help in simplifying the complex fraction: \[ \frac{(1 + e^{i\theta})e^{i\theta}}{(1 + e^{-i\theta})e^{i\theta}} = \frac{e^{i\theta} + e^{2i\theta}}{e^{i\theta} + 1}. \]

Step 6: Further Simplification

Since \(e^{2i\theta} = (e^{i\theta})^2\), we can factor out \(e^{i\theta}\) in the numerator: \[ \frac{e^{i\theta}(1 + e^{i\theta})}{e^{i\theta} + 1}. \]

Step 7: Canceling Common Terms

Cancel out the common term \(1 + e^{i\theta}\) from the numerator and the denominator: \[ e^{i\theta}. \]

Step 8: Interpreting the Result

The final result is \(e^{i\theta}\), which implies the argument of \(\left(\frac{1+z}{1+\bar{z}}\right)\) is \(\theta\). Therefore, the correct answer is option (B) \(\theta\).

Key Concepts

Unit Modulus

When we talk about a complex number having a unit modulus, it's quite simple to visualize. The term 'modulus' refers to the absolute value or the distance of the complex number from the origin in the complex plane. A complex number with a unit modulus has a modulus of 1. This means it lies on the unit circle centered at the origin.

In mathematical terms, if a complex number is represented as \(z = a + bi\), where \(a\) and \(b\) are real numbers, its modulus is given by \(|z| = \sqrt{a^2 + b^2}\.\) For it to have a unit modulus, this simplifies to:

• \(|z| = 1\)
• \( \sqrt{a^2 + b^2} = 1\), which implies \(a^2 + b^2 = 1\)

With a unit modulus, a complex number has a special form. If expressed in polar coordinates, it can be written as \(z = e^{i\theta}\), where \(\theta\) is the argument of the complex number. Thus, understanding unit modulus is essential in interpreting complex numbers on a unit circle.

Argument of a Complex Number

The argument of a complex number is a crucial part of understanding complex numbers. It represents the angle that the complex number, when plotted in the complex plane, makes with the positive direction of the real axis. This angle is typically measured in radians.

Given a complex number \(z = x + yi\), the argument \(\text{arg}(z)\) can be determined using the tangent function:

• \(\tan(\theta) = \frac{y}{x}\)
• The angle \(\theta\) is what we call the argument

In polar form, a complex number \(z\) is expressed as \(z = r\cdot e^{i\theta}\), where \(r\) is the modulus of the complex number, and \(\theta\) is the argument.

The argument helps us to locate a complex number on the complex plane uniquely using just the modulus and the argument. It's like the direction in which the complex number 'points' from the origin.

Complex Conjugate

Understanding the concept of the complex conjugate is essential when dealing with operations involving complex numbers. If a complex number is written as \(z = a + bi\), its complex conjugate, noted as \(\bar{z}\), is the number \(a - bi\). Essentially, we switch the sign of the imaginary part.

The complex conjugate is useful because it has particular properties that help simplify many complex number calculations:

• Multiplying a complex number by its conjugate results in a real number:
• \(z \cdot \bar{z} = (a + bi)(a - bi) = a^2 + b^2\)
• This real number is the square of the modulus of \(z\)

For a complex number with unit modulus, such as \(z = e^{i\theta}\), the complex conjugate is \(\bar{z} = e^{-i\theta}\). This conjugate simplifies expressions and is crucial in dividing complex numbers or finding reciprocal complex numbers, as seen in problems like the one provided.