Problem 199 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\left|z^{2}-1\right|=|z|^{2}+1\), then \(z\) lies on \(\quad\) [2004] (A) the real axis (B) an ellipse (C) a circle (D) the imaginary axis.

Step-by-Step Solution

Answer

The solution indicates that \( z \) lies on the real axis, thus option (A) is correct.

Step 1: Understanding the Given Equation

We are given the equation \( \left|z^{2} - 1\right| = |z|^{2} + 1 \). This equation relates the modulus of a complex number \( z \). We need to explore what conditions on \( z \) satisfy this equation.

Step 2: Expressing "z" in Terms of a Complex Number

Let \( z = x + yi \), where \( x \) and \( y \) are real numbers, and \( i = \sqrt{-1} \). The modulus \( |z| \) becomes \( \sqrt{x^2 + y^2} \). We need to substitute these expressions into the given equation.

Step 3: Substitute and Simplify Expression

Substitute \( z = x + yi \) into the equation: \( |z^2 - 1| = |z|^2 + 1 \).- Calculate \( z^2 = (x + yi)^2 = x^2 - y^2 + 2xyi \). - Thus, \( z^2 - 1 = x^2 - y^2 - 1 + 2xyi \).- So, \( \left|z^2 - 1\right| = \sqrt{(x^2 - y^2 - 1)^2 + (2xy)^2} \).The given equation becomes:\[ \sqrt{(x^2 - y^2 - 1)^2 + (2xy)^2} = x^2 + y^2 + 1 \]

Step 4: Analyze the Equation Further

Square both sides to eliminate the square root:\[(x^2 - y^2 - 1)^2 + (2xy)^2 = (x^2 + y^2 + 1)^2 \]Expand both sides:- LHS: \((x^2 - y^2 - 1)^2 + 4x^2y^2 = (x^4 - 2x^2y^2 + y^4 - 2x^2 + 2y^2 + 1) + 4x^2y^2 \)- RHS: \(x^4 + 2x^2y^2 + y^4 + 2x^2 + 2y^2 + 1 \)Combine terms and simplify. Check if the given condition holds for any specific set of values (such as when \(y = 0 \)).

Step 5: Simplification and Conclusion

Upon further simplification, both LHS and RHS reduce to the same expression when \( y = 0 \), giving:\( x^4 - 2x^2 + 1 = x^4 + 1 + 2(x^2) \)Thus, the terms involving \( y \) cancel, and we confirm \( y^2 = 0 \), hence \( y = 0 \).This implies \( z \) lies along the real axis. The final conclusion: the solution satisfies the condition of lying on the real axis.

Key Concepts

Modulus of Complex Numbers

In the world of complex numbers, understanding the concept of modulus is crucial. Simply put, the modulus of a complex number is its absolute value. Think of it as the distance from the origin to the point representing the complex number on the complex plane. When a complex number is expressed as \( z = x + yi \), where \( x \) and \( y \) are real numbers, the modulus \( |z| \) is found using the formula \( |z| = \sqrt{x^2 + y^2} \). Consider the modulus as a measure of magnitude or size, without regard to direction or angle. For instance:

• If the complex number is \( 3 + 4i \), its modulus is \( \sqrt{3^2 + 4^2} = 5 \).
• This value gives a sense of how far the point is from the origin in the geometric representation on the complex plane.

In many problems, like the one at hand, calculating and comparing the moduli of complex expressions reveals symmetries or conditions satisfied by the complex number \( z \). Here, applying the modulus helps identify that the solution involves simplicity on the real axis.

Real Axis

The real axis is a fundamental concept in complex number analysis. It represents all the complex numbers whose imaginary part is zero (i.e., \( y = 0 \)). On the complex plane, this aligns with the horizontal axis, showing only the real numbers without any imaginary component. Points on the real axis have forms such as:

• \( z = x + 0i \) or simply \( z = x \).
• Here, the modulus is identical to the absolute value of the real part: \( |z| = |x| \).

In the exercise solution, analyzing where \( z \) lies on the real axis simplifies the original equation. This solution reveals that when \( y = 0 \), the equation holds true. Thus, ensuring that the modulus condition is satisfied along this axis.

Complex Plane Geometry

Visualizing complex numbers on the complex plane enhances the understanding of their geometric properties. This plane consists of two axes: the real axis and the imaginary axis. Each complex number \( z = x + yi \) corresponds to a unique point in this plane, with \( x \) specifying the horizontal position (real part) and \( y \) indicating the vertical position (imaginary part). Complex plane geometry shows how transformations and operations on complex numbers translate into movements and patterns. For instance:

• Adding complex numbers translates to vector addition in the plane, which shifts points accordingly.
• In our problem, transformations reframe when points lie on specific loci, such as circles or lines.

Furthermore, it provides clarity in understanding relationships such as modulus and argument of complex numbers. Recognizing that a complex number lies on the real axis, as justified in this solution, is straightforward when seen geometrically. The real axis is a line across the horizontal of the plane, capturing all points without imaginary components, simplifying the given equation.