Problem 67 - 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 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: Identify Condition for Real Expression

For the expression \( \frac{z^2}{z-1} \) to be real, its imaginary part must be zero. Let's write \( z \) as a complex number \( z = x + yi \), where \( x \) and \( y \) are real numbers.

Step 2: Substitute Complex Number

Substitute \( z = x + yi \) into \( \frac{z^2}{z-1} \): \[ \frac{z^2}{z-1} = \frac{(x + yi)^2}{(x - 1) + yi} = \frac{(x^2 - y^2 + 2xyi)}{(x - 1) + yi} \]

Step 3: Calculate Real and Imaginary Parts

Calculate the real and imaginary parts by multiplying the numerator and denominator by the conjugate of the denominator: \[ \frac{(x^2 - y^2 + 2xyi)((x - 1) - yi)}{((x - 1)^2 + y^2)}. \] Expand and simplify to find expressions for the real and imaginary parts.

Step 4: Set Imaginary Part to Zero

To ensure that the expression is real, set the imaginary part obtained in the previous step to zero. This gives the condition:\[ 2xy(x - 1) + y(x^2 - y^2) = 0 \] which can be simplified to:\[ y(x^2 - y^2 + 2x - 2) = 0 \].

Step 5: Solve for Conditions: y=0 or Circle

The equation \( y(x^2 - y^2 + 2x - 2) = 0 \) implies either \( y = 0 \) or \( x^2 - y^2 + 2x - 2 = 0 \): - \( y = 0 \) means the point is on the real axis.- The equation \( x^2 - y^2 + 2x - 2 = 0 \) represents a circle which can be rewritten as \( (x+1)^2 + y^2 = 3 \).

Step 6: Conclusion on Point's Location

Therefore, the complex number \( z \) can either lie on the real axis or on the circle represented by \( (x+1)^2 + y^2 = 3 \), which passes through the origin due to being transformed from \( x^2 - y^2 + 2x - 2 = 0 \). Thus, the correct answer is (A).

Key Concepts

Real and Imaginary Parts

Every complex number has a real part and an imaginary part. A complex number is expressed as \( z = x + yi \), where \( x \) is the real part, and \( y \) is the imaginary part. Understanding these components is essential in solving problems involving complex numbers because they allow us to break down and examine each part separately.

When working with expressions like \( \frac{z^2}{z-1} \), finding the real and imaginary parts helps us determine if the expression is purely real. For an expression to be considered real, the imaginary part must equal zero. In this scenario, we substitute \( z = x + yi \) and expand it to separate the real part from the imaginary part. This technique is often used in many complex number problems to find the conditions under which a given expression can be purely real.

Complex Number Representation

Complex numbers can be represented in different forms, the most common being the Cartesian form, \( z = x + yi \). This form clearly shows the real part \(x\) and the imaginary part \(yi\). Understanding this representation is crucial in transitioning to other forms, such as polar and exponential representations, although the Cartesian form is more useful for algebraic operations like addition and multiplication.

In problems involving complex numbers, representing \( z \) in the \( x + yi \) form allows us to easily visualize its position on the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This visual representation can simplify the process of solving complex problems by translating abstract numerical values into geometric shapes and lines.

Geometric Interpretation of Complex Numbers

Geometrically, every complex number \( z = x + yi \) can be represented as a point \((x, y)\) on the complex plane. This plane is 2-dimensional, with one axis for the real part and another for the imaginary part.

In the exercise, the geometric interpretation helps in understanding the possible locations of the point represented by the complex number. The expression \((x+1)^2 + y^2 = 3\) from the exercise translates into a circle on the complex plane with a center at \((-1, 0)\) and a radius of \(\sqrt{3}\).

This circle passes through the origin, meaning any point \((x, y)\) on this circle satisfies the equation. Additionally, when \( y = 0 \), the complex number is purely real, and the point lies on the real axis. A geometric interpretation like this highlights the relationship between the algebraic expression of a complex number and its graphical representation, thus aiding in visualizing how complex numbers behave under certain conditions.