Problem 212 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(z \neq 1\) and \(\frac{z^{2}}{z-1}\) is real, then the point which is represented by the complex number \(z\) lies (A) either on the real axis or on a circle passing through the origin (B) on a circle with centre at the origin (C) either on the real axis or on a circle not passing through the origin (D) on the imaginary axis

Step-by-Step Solution

Answer

A: Either on the real axis or on a circle passing through the origin.

Step 1: Understand the Problem

We have a complex number \( z \) such that \( z eq 1 \) and the expression \( \frac{z^2}{z-1} \) is real. We need to determine where the point represented by \( z \) lies in the complex plane.

Step 2: Consider the Real Condition

The condition that \( \frac{z^2}{z-1} \) is real implies the expression has no imaginary part. Let \( z = x + yi \), where \( x \) and \( y \) are real numbers and \( i \) is the imaginary unit. The expression becomes \( \frac{(x + yi)^2}{x + yi - 1} = \frac{x^2 - y^2 + 2xyi}{x - 1 + yi} \).

Step 3: Simplify the Expression

Simplify \( \frac{z^2}{z-1} = \frac{x^2 - y^2 + 2xyi}{x - 1 + yi} \) by multiplying numerator and denominator by the conjugate of the denominator, \( x - 1 - yi \). This results in a real part \( \frac{(x^2 - y^2)(x-1) + 2xy^2}{(x-1)^2 + y^2} \) and an imaginary part \( \frac{2xy(x-1) - y(x^2-y^2)}{(x-1)^2 + y^2} \).

Step 4: Set Imaginary Part to Zero

For the expression to be real, the imaginary part must be zero: \[ 2xy(x-1) - y(x^2 - y^2) = 0 \]. Factor out \( y \): \[ y(2x(x-1) - (x^2 - y^2)) = 0 \]. If \( y = 0 \), \( z \) is on the real axis; otherwise, solve \( 2x(x-1) = x^2 - y^2 \).

Step 5: Solve the Equation

From \( 2x(x-1) = x^2 - y^2 \), we simplify to \( 2x^2 - 2x = x^2 - y^2 \) leading to \( x^2 + y^2 = 2x \). Rearranging gives a circle equation: \[ (x-1)^2 + y^2 = 1 \].

Step 6: Conclusion

The point \( z \) lies either on the real axis \( y = 0 \) or on the circle \( (x-1)^2 + y^2 = 1 \). The circle passes through the origin when \( x = 0 \), therefore option A is correct.

Key Concepts

Complex Plane

The complex plane, also known as the Argand plane, is a visual tool used to represent complex numbers. In this plane:

• The horizontal axis, or the x-axis, represents the real part of complex numbers.
• The vertical axis, or the y-axis, represents the imaginary part.

Complex numbers are written in the form of \( z = x + yi \), where \( x \) is the real part and \( y \) is the imaginary part. Each complex number corresponds to a unique point on this plane. The location of a point on the complex plane helps us to understand the relationship between its real and imaginary components.
Visualizing complex numbers with this method aids in performing arithmetic operations and finding relationships between different numbers.

Real Axis

The real axis is a crucial part of the complex plane. It serves as the baseline or reference point for measuring real values of complex numbers. Let's break it down:

• It coincides with the horizontal x-axis on the complex plane.
• Any point on the real axis has an imaginary part equal to zero, meaning it can be expressed as \( z = x \).

When dealing with problems involving complex numbers, the real axis helps determine where numbers like \( z = x \), with no imaginary component, are located. In our exercise, a condition that returns us to the real axis is when the imaginary component \( y \) equals zero.

Circle Equation

In mathematics, circles can also exist on the complex plane, where they represent collections of points that satisfy a circle equation. A typical circle equation in this plane looks like \[(x - h)^2 + (y - k)^2 = r^2,\]where \( (h, k) \) is the center, and \( r \) is the radius.
For example, in the given exercise, we ended with the circle equation \((x-1)^2 + y^2 = 1\).Here, the circle's center is at \( (1, 0) \), and the radius is 1. This circle notably passes through the origin, a point where both the real and imaginary parts are zero. The circle defines a boundary where solutions to our equation with complex numbers must lie.

Imaginary Unit

The imaginary unit \( i \) is a fundamental concept in understanding complex numbers. It is defined uniquely by its property that

• \( i^2 = -1 \).
• \( i \) helps in expanding the realm of real numbers to include solutions to equations like \( x^2 + 1 = 0 \), which are not solvable using only real numbers.

When a complex number is expressed in the form \( z = x + yi \), the imaginary unit \( i \) indicates that \( y \) is an imaginary part. In our solution steps, it plays a role in forming and simplifying expressions, helping isolate real and imaginary components as needed. Understanding \( i \) allows us to explore a world of numbers beyond the real number line alone.