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

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

Option (B): \(z_{1} \times z_{2}\) is unimodular.

Step 1: Understand Given Equation

We start with the equation \( \left|z_{1}+z_{2}\right|=\left|\frac{1}{z_{1}}+\frac{1}{z_{2}}\right| \). This can be rewritten as \( \left| z_1 + z_2 \right| = \left| \frac{z_1 + z_2}{z_1 z_2} \right| \).

Step 2: Simplify the Absolute Value Equation

By simplifying \( \left| z_1 + z_2 \right| = \frac{\left| z_1 + z_2 \right|}{\left| z_1 z_2 \right|} \), we see that it simplifies to \( \left| z_1 z_2 \right| = 1 \). This implies that \( z_1 \times z_2 \) is unimodular.

Key Concepts

Unimodular Complex Numbers

Unimodular complex numbers are an interesting type of complex number. They are defined as numbers whose absolute value is equal to 1. In simpler terms, if we take a complex number, say \( z \), and calculate its absolute value which is represented as \( |z| \), a unimodular complex number satisfies the equation \( |z| = 1 \).
Examples of unimodular complex numbers include \( 1, -1, i, \text{and} -i \). These are points located on the unit circle of the complex plane, meaning they are a specific distance (1 unit) away from the origin.
Unimodular numbers have a special property in multiplication. When two complex numbers multiply together to form a unimodular number, it usually hints at a unique and symmetrical relationship between the numbers, as seen in problems involving complex solutions.

Absolute Value of Complex Numbers

The absolute value of a complex number provides a measure of its magnitude or distance from the origin on the complex plane. For a complex number \( z = a + bi \), where \( a \) and \( b \) are real numbers, the absolute value \( |z| \) is calculated using the formula: \[ |z| = \sqrt{a^2 + b^2} \]
This is similar to the Pythagorean theorem where \( a \) and \( b \) are sides of a right triangle, and \( |z| \) is the hypotenuse.
Knowing the absolute value helps determine how far a number is from zero and can be useful in many operations, such as when deciding if two sum or product results in unimodular behavior.
In the given exercise, the condition \( \left|z_{1} + z_{2}\right| = \left| \frac{1}{z_{1}} + \frac{1}{z_{2}} \right| \) becomes simplified to find \( |z_1 z_2| = 1 \). This indicates that the product \( z_1 \times z_2 \) is unimodular, deepening the importance of understanding absolute values in these problems.

Complex Number Multiplication

Complex number multiplication involves both magnitude and angle considerations. When two complex numbers \( z_1 \) and \( z_2 \) are multiplied, their magnitudes multiply, and their angles add up.
If \( z_1 = a + bi \) and \( z_2 = c + di \), their product is:\[ z_1 \times z_2 = (ac - bd) + (ad + bc)i \]
This operation not only combines the real and imaginary parts separately but also impacts the final result's magnitude and angle.
In the context of unimodular numbers, this product becomes particularly interesting. If the product \( |z_1 z_2| = 1 \), it implies that they have perfect balancing magnitudes that together, their unit-circle properties bring about a simplification feature, reflecting the unit modulus trait.
Understanding complex multiplication is crucial in identifying harmonious relationships in complex number operations which are frequent in many mathematical and engineering fields.