Problem 44 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(|z|=1\), then the value of \(\left(\frac{z-1}{z+1}\right)\) is (A) 0 (B) purely real (C) purely imaginary (D) complex number

Step-by-Step Solution

Answer

The value is purely imaginary (Option C).

Step 1: Understanding the Problem

The problem involves the modulus of a complex number \( z \), where \(|z| = 1\). We need to analyze the expression \( \frac{z-1}{z+1} \) to determine its nature (real, imaginary, or complex).

Step 2: Modulus Condition of Complex Number

The condition \(|z| = 1\) implies that the complex number \( z \) lies on the unit circle in the complex plane. This means that \( z = e^{i\theta} \) for some \( \theta \), making it a number of the form \( \cos \theta + i\sin \theta \).

Step 3: Calculation of Conjugate

For complex numbers, if \(|z| = 1\), then the conjugate \( \overline{z} \) is equal to \( \frac{1}{z} \). Hence, \( z \overline{z} = 1 \). Use this identity to simplify expressions.

Step 4: Simplifying the Expression

Consider the expression \( \frac{z-1}{z+1} \). Let's consider its conjugate: \( \overline{\left(\frac{z-1}{z+1}\right)} \).The conjugate of \(\frac{z-1}{z+1}\) becomes \(\frac{\overline{z} - 1}{\overline{z} + 1} = \frac{\frac{1}{z} - 1}{\frac{1}{z} + 1} = \frac{1 - z}{1 + z} \). This is \(-\frac{z-1}{z+1}\), proving it to be purely imaginary as dividing by complex conjugates inverts the sign.

Step 5: Determine the Nature of the Expression

Since \( \overline{\left(\frac{z-1}{z+1}\right)} = -\frac{z-1}{z+1} \), it follows that the expression \( \frac{z-1}{z+1} \) is purely imaginary. If \( \overline{w} = -w \), then \( w \) must be purely imaginary.

Key Concepts

Unit Circle

The unit circle is one of the most fundamental concepts when dealing with complex numbers. To grasp why it's essential, think about a circle centered at the origin of the complex plane (the axis consisting of real and imaginary units). A unit circle has a radius of one. Any point on this circle can be described by a complex number \( z \) with a modulus (or absolute value) of one, written as \(|z| = 1\).

Complex numbers on this circle are essential because:

• They can be expressed in the exponential form \( e^{i\theta} \), where \( \theta \) is the angle from the positive x-axis to the line joining the point to the origin.
• This form can also be broken down into \( \cos \theta + i\sin \theta \), utilizing Euler's formula.

The characteristics of the unit circle make computations with complex numbers more predictable. For instance, any multiplication involving numbers on this circle merely changes the orientation of the vector they represent, without altering its length.

Modulus Condition

The modulus condition for complex numbers specifies their magnitude relative to the origin in the complex plane. Specifically, for a complex number \( z = x + yi \), the modulus is represented as \( |z| = \sqrt{x^2 + y^2} \). The modulus condition \(|z| = 1\) implies that the point representing \( z \) lies somewhere on the unit circle.

Why is this condition significant? Here are a few reasons:

• When \(|z| = 1\), it effectively normalizes the complex number, ensuring its magnitude is constant.
• Complex numbers such as \( e^{i\theta} \) that satisfy this condition are called 'unit complex numbers'. They are often utilized in rotations and transformations.
• This condition ensures that operations involving such numbers, like multiplication or inversion, preserve properties due to their predictable magnitudes.

Hence, the condition \(|z| = 1\) sets a strong foundation for understanding the dynamics of complex expressions such as \( \frac{z-1}{z+1} \).

Conjugate

The concept of the conjugate of a complex number is a straightforward yet powerful tool in complex number theory. For a complex number \( z = a + bi \), its conjugate is \( \overline{z} = a - bi \). Conjugates play a pivotal role in many mathematical operations, especially when dealing with division in the complex domain.

Here are some key points to remember about conjugates:

• Conjugating alters the sign of the imaginary part while keeping the real part unchanged.
• For complex numbers \( z \) with modulus 1, \( \overline{z} = \frac{1}{z} \), because \( z \overline{z} = |z|^2 = 1 \).
• When simplifying expressions like \( \frac{z-1}{z+1} \), taking the conjugate of the expression often provides insights into the nature of the expression.

In the context of our original exercise, finding the conjugate helped demonstrate that \( \frac{z-1}{z+1} \) is purely imaginary. This is because its conjugate is just the negative of itself, a characteristic indicative of purely imaginary numbers.