Problem 39 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(z_{1} \neq-z_{2}\) and \(\left|z_{1}+z_{2}\right|=\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|\), then (A) at least one of \(z_{1}, z_{2}\) is unimodular (B) \(z_{1} \times z_{2}\) is unimodular (C) both \(z_{1}, z_{2}\) are unimodular (D) None of these

Step-by-Step Solution

Answer

(B) \(z_1 \times z_2\) is unimodular.

Step 1: Understand the Given Equation

We are given \(\left|z_{1}+z_{2}\right|=\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right|\). The expression \(\frac{1}{z_{1}}+\frac{1}{z_{2}}\) simplifies to \(\frac{z_1 + z_2}{z_1 z_2}\). Thus, the equation can be rewritten as \(\left|z_1 + z_2\right| = \left|\frac{z_1 + z_2}{z_1 z_2}\right|\).

Step 2: Analyze the Simplified Equation

Upon simplification, the equation \(\left|z_1 + z_2\right| = \left|\frac{z_1 + z_2}{z_1 z_2}\right|\) further simplifies to \(1 = \left|\frac{1}{z_1 z_2}\right|\). Therefore, \(|z_1 z_2| = 1\).

Step 3: Conclusion Based on Analysis

Since \(|z_1 z_2| = 1\), it implies that the product \(z_1 \times z_2\) is unimodular (has a modulus of 1). This corresponds to option (B).

Key Concepts

Modulus of a Complex Number

The modulus of a complex number is a crucial concept in understanding complex numbers. The modulus is the distance of the complex number from the origin when plotted on the complex plane. It is defined for a complex number \( z = a + bi \) as \( |z| = \sqrt{a^2 + b^2} \).
The modulus serves many purposes, providing a measure of the size, or magnitude, of the complex number.

• The modulus is always a non-negative real number.
• The modulus of zero is zero.
• The modulus can also be understood as the absolute value in the context of complex numbers.

Understanding the modulus is essential when exploring properties of complex numbers like complex multiplication and identifying unimodular numbers. In the given exercise, knowing how to manipulate the modulus allows one to verify whether a complex number or its combinations meet specific conditions.

Unimodular Complex Numbers

A unimodular complex number is simply a complex number that has a modulus of one. That means for a complex number \( z = a + bi \), it is unimodular if \( |z| = 1 \).
Unimodular numbers hold a significant position in complex number analysis due to their unique properties. For any unimodular number:

• Their amplitudes remain constant as the modulus is always one.
• These numbers lie on the unit circle in the complex plane.
• Unimodular numbers preserve magnitude in multiplication operations.

In the context of the exercise, the conclusion that \( z_1 \times z_2 \) is unimodular arises from the condition \( |z_1 z_2| = 1 \), showing the unique nature of unimodular numbers in complex multiplication.

Complex Multiplication

Complex multiplication involves combining two complex numbers to produce another complex number. When multiplying two complex numbers, say \( z_1 = a + bi \) and \( z_2 = c + di \), the product is given by:\[ z_1 \times z_2 = (ac - bd) + (ad + bc)i \]
Complex multiplication is not only about performing arithmetic; it reveals crucial properties:

• It combines both the real and imaginary parts of the numbers.
• The product of complex numbers reflects geometric transformations, such as rotations and scaling, on the complex plane.
• Multiplying unimodular complex numbers results in another unimodular number, preserving its position on the unit circle.

In the given problem, the result \( |z_1 z_2| = 1 \) illustrates a key characteristic of how complex numbers interact through multiplication, often involving modulus and lending to deeper insights into their collective behavior.