Problem 44 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(C) purely imaginary

Step 1: Understand the given condition

We are given that \(|z| = 1\), which means that the complex number \(z\) is on the unit circle in the complex plane. This implies that \(z\) can be expressed as \(z = e^{i\theta}\), where \(\theta\) is the argument of \(z\).

Step 2: Express the fraction

We need to evaluate \(\left(\frac{z-1}{z+1}\right)\). Substitute \(z = e^{i\theta}\) into the expression to get \(\frac{e^{i\theta} - 1}{e^{i\theta} + 1}\).

Step 3: Find the modulus of the expression

Using the property that \(|z|=1\), calculate the modulus \(|\frac{z-1}{z+1}| = \frac{|e^{i\theta} - 1|}{|e^{i\theta} + 1|}\). Simplifying gives \(\frac{|z-1|}{|z+1|}\).

Step 4: Analyze the behavior on the unit circle

For the expression to remain on the unit circle or exhibit specific properties (purely imaginary or purely real), the values in the counter and denominator must have particular relationships. In polar form, this often results in cancellation or trigonometric identities leading to imaginary or real values.

Step 5: Test the value of the expression

Consider \(z = -1\), then \(\frac{z-1}{z+1} = \frac{-1-1}{-1+1} = \frac{-2}{0}\), which is undefined. For \(z = i\), \(\frac{i-1}{i+1}\) and check for various cases to explore imaginary possibilities.

Step 6: Conclude

Set \(z = e^{i\theta}\), dividing by \(z+1\) leads to calculations resulting often in values \( \cos + i \sin \), which lean toward imaginary expansions with zero real part for particular angles. This stems from trigonometric interpretations and symmetries of \(e^{i\theta}\) when divided and pattern resulting from differences.

Key Concepts

Unit Circle

In mathematics, the Unit Circle is a circle of radius 1 centered at the origin of the complex plane. The significance of the unit circle in the context of complex numbers is that any complex number on this circle has a modulus of 1. This is crucial when dealing with expressions involving complex numbers, such as

• Evaluating expressions like \(|z| = 1\), which indicates that the complex number \(z\) is on the unit circle.
• Expressing complex numbers in polar form as \(z = e^{i\theta}\), where \(\theta\) is the angle formed with the positive real axis.

When \(z\) is on the unit circle, this also implies that the complex number can be expressed using Euler's formula, where \(z = \cos\theta + i\sin\theta\). This representation makes it easier to analyze and manipulate complex numbers geometrically.

Modulus

The modulus of a complex number, often represented as \(|z|\), is a measure of its distance from the origin in the complex plane. It is calculated as \(|z| = \sqrt{x^2 + y^2}\), where \(x\) and \(y\) are the real and imaginary parts of the complex number \(z = x + yi\).

• For unit circle calculations, this modulus is always 1.
• This means any transformation or manipulation of complex numbers indexed on the unit circle must retain this modulus.

In the context of the exercise provided, the modulus helps in simplifying complex expressions. For example, when calculating the modulus of \(\left(\frac{z-1}{z+1}\right)\), using the fact that \(|z| = 1\) helps verify consistency and characteristics, such as whether the expression is purely imaginary or real.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. They are essential tools in simplifying expressions involving unit circle and complex numbers. Some of the common identities include:

• \(\cos^2\theta + \sin^2\theta = 1\), which relates directly to the unit circle.
• \(\tan\theta = \frac{\sin\theta}{\cos\theta}\).

These identities become even more powerful when dealing with complex numbers on the unit circle. They allow expressions using \(e^{i\theta}\) to be rewritten in terms of sine and cosine. In solving the given problem, when \(z\) is expressed as \(e^{i\theta}\), trigonometric identities help simplify calculations and verify if the resulting expression is purely imaginary. By understanding the role of these trigonometric identities, one can navigate complex expressions to quickly assess real versus imaginary components.