Problem 77 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(|z+\bar{z}|+|z-\bar{z}|=8\), then \(z\) lies on (A) a circle (B) a straight line (C) a square (D) None of these

Step-by-Step Solution

Answer

(C) a square

Step 1: Understand the terms

The expression involves complex numbers, where \(z = x + yi\) for real numbers \(x\) and \(y\), and \(\bar{z} = x - yi\) is the conjugate of \(z\). The modulus \(|z|\) or \(|\bar{z}|\) represents the magnitude of the complex number.

Step 2: Simplify the expression

Calculate the real and imaginary parts of the expressions \(z+\bar{z}\) and \(z-\bar{z}\). We obtain:\[z + \bar{z} = 2x\]\[z - \bar{z} = 2yi\]Therefore, the moduli are: \[|z+\bar{z}| = |2x|\] \[|z-\bar{z}| = |2yi| = 2|y|\]Substitute these in the given equation: \[|2x| + 2|y| = 8\]

Step 3: Solve the modulus equation

Divide the entire equation by 2 to simplify:\[|x| + |y| = 4\]This is the equation for the equation of a square centered at the origin in the coordinate plane, rotated by 45 degrees.

Step 4: Interpret the solution geometrically

The equation \(|x| + |y| = 4\) represents the boundary of a diamond shape or square with vertices at (4,0), (0,4), (-4,0), and (0,-4) in the coordinate plane. This implies that the set of points for \(z\) with these conditions forms a square.

Key Concepts

Modulus of Complex Numbers

Understanding the modulus of complex numbers is key to mastering their properties and applications. A complex number often takes the form:

• \( z = x + yi \)

where \(x\) and \(y\) are real numbers. The modulus, or absolute value, of \(z\), is a measure of its distance from the origin in the complex plane.
This is calculated as:

• \( |z| = \sqrt{x^2 + y^2} \)

The modulus provides insight into the magnitude of the number without considering its direction.
For example, in the exercise, \(|z+\bar{z}|\) was found to be \(|2x|\), reflecting the influence of the real component of the sum. The modulus is always a non-negative value and offers crucial information when solving or interpreting complex equations.

Complex Conjugate

The concept of a complex conjugate is essential in complex number theory. For a complex number \(z = x + yi\), its conjugate, denoted as \(\bar{z}\), is given by:

• \( \bar{z} = x - yi \)

This effectively mirrors the complex number across the real axis in the complex plane.
This operation is particularly useful as it eliminates the imaginary component when adding or multiplying a complex number by its conjugate.
In the context of the exercise, manipulating \(z\) and \(\bar{z}\) allowed us to express the complex numbers in terms of their real and imaginary parts, facilitating an easier solution.
The expression \(z - \bar{z} = 2yi\) focuses solely on the imaginary part and plays a vital role in geometric interpretation and solving complex equations.

Geometric Interpretation of Complex Equations

Understanding the geometric interpretation of complex equations involves visualizing complex numbers in a two-dimensional plane, known as the complex plane. Here, the real part of a complex number \(x + yi\) is plotted along the x-axis, while the imaginary part is plotted along the y-axis.
This approach assists in solving the exercise equation:

• \(|x| + |y| = 4\)

This results in the coordinates forming a square rotated by 45 degrees, with vertices at (4,0), (0,4), (-4,0), and (0,-4).
Such geometric interpretations facilitate understanding the locus of solutions that satisfy given complex conditions.
Visualizing the geometry of complex numbers simplifies the analysis and solution of equations that might otherwise appear abstract or challenging.