Problem 211 - A Complete Resource Book in Mathematics for JEE Main

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 [2012] (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 Exercise

We need to determine where the complex number \(z\) lies such that the expression \(\frac{z^2}{z-1}\) is real, given that \(z eq 1\).

Step 2: Express the Complex Number

Let \(z = x + yi\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit. Substitute this into the expression \(\frac{z^2}{z-1}\).

Step 3: Substitute and Simplify

Substitute \(z = x + yi\) into \(z^2\) to get \((x + yi)^2 = x^2 - y^2 + 2xyi\). For \(z-1\), substitute to get \((x-1) + yi\). Therefore, \(\frac{z^2}{z-1} = \frac{x^2 - y^2 + 2xyi}{(x-1) + yi}\).

Step 4: Multiply by the Conjugate

To make the function real, multiply numerator and denominator by conjugate of \((x-1) + yi\), which is \((x-1) - yi\). This simplifies to \(\frac{(x^2 - y^2 + 2xyi)((x-1) - yi)}{((x-1)^2 + y^2)}\).

Step 5: Simplify and Find Real Part

The expression can be simplified to get a real part if the imaginary part of the numerator is zero. Simplify \((x^2 - y^2 + 2xyi)((x-1) - yi)\) and equate the imaginary component to zero.

Step 6: Equate Imaginary Part to Zero

The imaginary component from multiplication yields: \((x^2 - y^2)(-y) + 2xy(x-1) = 0\). Solve this to find points on the complex plane.

Step 7: Identify Condition for Real Part

Solve: \((x^2 - y^2)y = 2x^2y - 2xy = 0\). This implies either \(y = 0\) (real axis) or \(x^2 - x - y^2 = 0\) (a circle through the origin).

Step 8: Conclusion on the Type of Circle

The circle equation derived \(x^2 - x - y^2 = 0\) is a circle that passes through the origin. Hence, \(z\) is either on the real axis or the described circle.

Key Concepts

Real and Imaginary Parts

In the world of complex numbers, each number involves two parts: the real part and the imaginary part. A complex number is generally expressed as \(z = x + yi\), where \(x\) is the real part, and \(yi\) is the imaginary part. Here, \(x\) and \(y\) are real numbers, and \(i\) represents the imaginary unit, with the key property that \(i^2 = -1\).
Real numbers lie on the horizontal axis, and imaginary numbers are placed on the vertical axis of the complex plane. By thinking about complex numbers in this way, we can effectively take advantage of both the algebraic and geometric representations.

• Real Part: Corresponds to the value on the horizontal axis.
• Imaginary Part: Related to the vertical axis, scaled by \(i\).

In exercises involving complex numbers, often the goal is to manipulate these parts to understand their geometric implication or make an expression purely real or imaginary.

Complex Plane

The complex plane is a visual tool used for representing complex numbers. Think of it like a 2D graph where each point corresponds to a complex number. It consists of a horizontal axis, known as the real axis, and a vertical axis, called the imaginary axis. A complex number \(z = x + yi\) is represented as a point \((x, y)\) in this plane.

• Real Axis: Represents all real numbers, lying horizontally.
• Imaginary Axis: Represents all imaginary numbers, aligned vertically.

By plotting complex numbers on this plane, we can easily visualize their interactions and transformations. For instance, the real component determines the position along the horizontal axis, while the imaginary component sets the vertical position.
This duality helps us to interpret complex arithmetic as geometrical transformations, such as rotation and scaling, which can be vital in understanding complex equations' solutions, as demonstrated in exercises like determining when \(\frac{z^2}{z-1}\) becomes a real number.

Geometric Representation of Complex Numbers

Complex numbers don't just have algebraic expressions; they can also be understood geometrically. A complex number \(z = x + yi\) can be viewed as a vector from the origin to the point \((x, y)\) in the complex plane. This geometrical perspective allows us to interpret operations on complex numbers in ways that are visually intuitive.

• Magnitude (Modulus): This is the distance from the origin to the point, calculated as \(\sqrt{x^2 + y^2}\).
• Argument (Angle): The angle \(\theta\) between the vector and the real axis, usually measured counterclockwise.

For example, when a complex number defines a circle in the plane, it involves a set of points equidistant from a particular center. In this exercise, solving the condition \(x^2 - x - y^2 = 0\) geometrically indicates a circle passing through the origin. This geometry helps us conclude that the position of \(z\) is either along the real axis or on this specific circle.