Problem 45 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(z_{1}\) and \(z_{2}\) are complex numbers, such that \(z_{1}+z_{2}\) is a real number, then (A) \(z_{1}=-\bar{z}_{2}\) (B) \(z_{2}=\bar{z}_{1}\) (C) \(z_{1}\) and \(z_{2}\) are any two complex numbers (D) \(z_{1}=\bar{z}_{1}, z_{2}=\bar{z}_{2}\)

Step-by-Step Solution

Answer

The correct option is (A) \( z_1 = -\bar{z}_2 \).

Step 1: Understand the given condition

We are given that the sum of two complex numbers \( z_1 + z_2 \) is a real number. A complex number is represented as \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with \( i^2 = -1 \). If a complex number is real, its imaginary part must be zero.

Step 2: Express complex numbers in terms of their real and imaginary parts

Let \( z_1 = a_1 + b_1 i \) and \( z_2 = a_2 + b_2 i \). Here, \( a_1, b_1, a_2, \) and \( b_2 \) are real numbers.

Step 3: Apply the condition that their sum is real

The sum is \( z_1 + z_2 = (a_1 + a_2) + (b_1 + b_2)i \). For this sum to be real, the imaginary part \( b_1 + b_2 \) must be zero, so \( b_1 + b_2 = 0 \).

Step 4: Relate the equations to the options

Since \( b_1 + b_2 = 0 \), we have \( b_1 = -b_2 \). This suggests \( z_1 = a_1 + b_1 i \) and \( z_2 = a_2 - b_1 i \). This implies that \( z_1 = \overline{-z_2} \), which is equivalent to \( z_1 = -\bar{z}_2 \).

Step 5: Verify the correct option

From the step above, option (A) is consistent with the derived equation \( z_1 = -\bar{z}_2 \), which satisfies the condition given.

Key Concepts

Real and Imaginary Parts

In the world of complex numbers, every complex number is made up of two components: the real part and the imaginary part. A complex number is typically written in the form \( z = a + bi \).

Here, \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the imaginary unit with the crucial property that \( i^2 = -1 \).

This representation helps in simplifying complex arithmetic by allowing us to separate the number into its components.

For example, if you have the complex number \( 3 + 4i \), then the real part is 3 and the imaginary part is 4.

• The real part is what you would see in regular real number arithmetic.
• The imaginary part determines the position on the imaginary axis.

In many mathematical applications, understanding how to separate and manipulate these parts is fundamental.

Conjugate of Complex Numbers

The conjugate of a complex number is a vital concept when dealing with complex calculations. The conjugate of a complex number \( z = a + bi \) is written as \( \bar{z} = a - bi \).

This operation essentially flips the sign of the imaginary part.

The conjugate is particularly useful because it can help simplify division within complex numbers and is crucial in many algebraic manipulations.

To understand better, let's consider \( z = 2 + 3i \). The conjugate is \( \bar{z} = 2 - 3i \).

• When you multiply a complex number by its conjugate, the result is always a real number; specifically, \( z \cdot \bar{z} = a^2 + b^2 \).
• Conjugates are also integral in recognizing and solving problems where the imaginary parts cancel each other, yielding a real result.

In the context of the original exercise, recognizing that \( z_1 = -\bar{z}_2 \) employs the property of conjugates to ensure the sum \( z_1 + z_2 \) remains real.

JEE Main Mathematics

Complex numbers are an important topic in competitive exams like JEE Main, where the syllabus covers their properties extensively.

Understanding real and imaginary parts, the use of conjugates, and the algebraic manipulation of complex numbers is crucial for solving related problems.

These concepts are woven into various question types, from simple arithmetic operations to complex number equations.

• Mastering these topics is typically seen as essential for building a foundation in both higher mathematics and engineering fields.
• The problems often involve applying the properties of complex numbers, like the condition that the sum of two numbers is real, as seen in our exercise.
• In preparation for exams like JEE Main, students should aim to be comfortable with both theoretical and practical applications of these concepts.

Understanding these ideas deeply not only aids in solving exam questions but is also a valuable tool in more advanced studies and practical applications.