Problem 67 - 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 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 Condition

The expression \(\frac{z^2}{z-1}\) is given as real, indicating that its imaginary part is zero. This implies that \(\text{Im}\left(\frac{z^2}{z-1}\right) = 0\). We must manipulate this condition further to understand where the complex number \(z\) lies in the complex plane.

Step 2: Set up the Complex Number

Let \(z = x + yi\) where \(x\) and \(y\) are real numbers. Then \(z^2 = (x + yi)^2 = x^2 - y^2 + 2xyi\) and \(z - 1 = x - 1 + yi\). The fraction is \(\frac{z^2}{z-1} = \frac{x^2 - y^2 + 2xyi}{x - 1 + yi}\).

Step 3: Simplify the Fraction

Multiply the numerator and denominator by the conjugate of the denominator to eliminate the imaginary part: \((x - 1 - yi)\). This leads to \(\frac{(x^2 - y^2 + 2xyi)(x - 1 - yi)}{(x-1)^2 + y^2}\).

Step 4: Calculate the Imaginary Part

After expanding and simplifying the terms, the imaginary part of the complex number \((x^2 - y^2 + 2xyi)(x - 1 - yi)\) should equal zero. This condition provides a relationship between \(x\) and \(y\).

Step 5: Analyze the Result

By setting the imaginary part to zero, analyze the expression and identify the set of \(z\) values satisfying this condition. The relation derived is \(y (x^2 + y^2 - x) = 0\).

Step 6: Solve for Possible Values of z

From \(y (x^2 + y^2 - x) = 0\), either \(y = 0\) (implying \(z\) lies on the real axis) or \(x^2 + y^2 = x\) (which is the equation of a circle centered at \((0.5, 0)\) with radius \(0.5\)).

Step 7: Final Condition Verification

This circle passes through the origin because when \(x = 0, y=0\), the equation \(x^2 + y^2 = x\) is satisfied. Thus, \(z\) lies either on the real axis or on a circle passing through the origin.

Key Concepts

Real Number Conditions

When we talk about real numbers in terms of complex numbers, we focus on expressions where the imaginary part equals zero. For example, an expression like \(\text{Im}\left(\frac{z^2}{z-1}\right) = 0\) tells us that the result has no imaginary component.
This provides critical information:

• If a fraction involving a complex number \(z\) is real, any imaginary terms must cancel each other out.
• The complex number \(z\) can be expressed as \(x + yi\), where \(x\) and \(y\) are real numbers, and \(i\) represents the square root of -1.

To make an expression real, you can perform operations like multiplying by the conjugate to remove imaginary parts. By manipulating these expressions, students explore how complex numbers relate to each other in cases constrained by real number conditions.

Equation of a Circle

The equation of a circle is a useful concept when dealing with complex numbers. Particularly, when the expression related to a complex number meets certain conditions, it translates into a geometric shape on the complex plane.
When you solve the equation \(x^2 + y^2 = x\), it represents a circle:

• With a center at \((0.5, 0)\)
• And a radius of 0.5

This circle equation means that every point \(z = x + yi\) adhering to this condition will lie on the circle defined by these parameters. It adds a visual interpretation of a previously abstract mathematical condition.By connecting these algebraic terms with geometric shapes, complex number problems can often be visualized and understood more intuitively.

Imaginary Part

In complex number expressions, the imaginary part plays a key role in understanding the nature of the number. The imaginary part arises when a complex number is expressed as \(a + bi\), where \(b\) is the imaginary component.
For the expression \(\frac{z^2}{z-1}\) to be real, its imaginary part needs to be zero. Simplifying the complex expression requires multiplying the numerator and denominator by the conjugate to eliminate \(i\).
Evaluating the imaginary part of an expanded expression reveals which components must zero out for the entire expression to be purely real or imaginary. This kind of analysis:

• Helps determine the condition of the equation relative to real numbers.
• Offers insight into the potential locations of \(z\) in the complex plane, such as lying on an axis or circle.

Mastering how imaginary components affect complex numbers allows students to solve complex equations and understand their geometric implications in the complex plane.