Problem 211 - 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) The point lies either on the real axis or on a circle passing through the origin.

Step 1: Understanding the Given Condition

We need to find where the complex number \(z\) lies if \(\frac{z^2}{z-1}\) is real and \(z eq 1\). This means that the imaginary part of the expression must equal zero.

Step 2: Expressing the Complex Number

Let \(z = x + yi\), where \(x\) and \(y\) are real numbers. This gives \(z^2 = (x + yi)^2 = x^2 - y^2 + 2xyi\).

Step 3: Calculating the Denominator and the Complex Expression

For \(z - 1 = (x - 1) + yi\), we compute \(\frac{z^2}{z-1} = \frac{x^2 - y^2 + 2xyi}{(x-1) + yi}\). Multiply the numerator and the denominator by the conjugate of the denominator to make the denominator real, \((x-1) - yi\).

Step 4: Simplifying to Obtain a Real Number Condition

This results in:\[\frac{(x^2 - y^2 + 2xyi)((x-1) - yi)}{(x-1)^2+y^2}\]Expanding and simplifying, we obtain:\[\frac{(x^2 - y^2)(x-1) + y^2 i - 2x_1y^2 i - 2xyy i}{(x-1)^2 + y^2}\]The imaginary part must be zero for the whole expression to be real.

Step 5: Constraint Satisfaction for Real Result

Set imaginary part to zero: \[-y(x^2 - y^2 - x + 2xy) = 0\]This resolves to either \(y = 0\) (lies on the real axis) or \(x^2 - y^2 = x - 2xy\), which is the equation of a circle.

Step 6: Analyzing Equation of Circle

Rearranging, \(x^2 - x = y^2 + 2xy\) resembles a circle equation. Completing the square for a circle, we get:\[x^2 - x + \frac{1}{4} + y^2 + 2iy = \text{constant} \]Upon solving this depicts a circle passing through the origin.

Key Concepts

Real Part

Complex numbers are expressed in the form \(z = x + yi\), where \(x\) and \(y\) are real numbers, and \(i\) is the imaginary unit. In this form, \(x\) is known as the real part. The real part of a complex number can be thought of as its horizontal coordinate on the complex plane.

• If a complex number has a zero imaginary part, it lies entirely on the real axis of the complex plane.
• In exercises involving real results, ensure the imaginary component vanishes to isolate the real part.

For instance, if \(z\) is such that \(\frac{z^2}{z-1}\) is real, the imaginary part must be zero, indicating a condition solely dependent on its real part.

Imaginary Part

The imaginary part of a complex number \(z = x + yi\) is \(y\). This part, represented with the unit \(i\), is essential for plotting a complex number in the vertical direction on the complex plane. The imaginary part influences whether a complex number exits off the real axis.

• To ensure that a mathematical expression is a real number, the imaginary component must equal zero.
• A zero imaginary part means the complex number sits either directly on the real axis or meets specific conditions if part of a more complex expression.

In our exercise, the condition \(\text{Imaginary part of } \frac{z^2}{z-1} = 0\) was essential for pinpointing \(z's\) location.

Circle Equation

A circle in the complex plane is represented by a particular set of conditions applied to a complex number \(z\). Typically, a complex circle equation can take the form \((x - a)^2 + (y - b)^2 = r^2\), where \(a\) and \(b\) are coordinates of the circle's center, and \(r\) is the radius.

• Understanding how to derive or recognize circle equations helps visualize complex numbers beyond linear alignments on the real or imaginary axes.
• The condition \(x^2 - y^2 = x - 2xy\) simplifies to a circle equation that passes through designated points like the origin.

During the exercise, analysis of constraints led to forming this circle equation, indicating the potential path or locus that \(z\) can trace.

Completing the Square

Completing the square is a mathematical method used to transform a quadratic equation into a convenient square form. This process is particularly useful for solving quadratic equations and recognizing conic sections like circles.

• To complete the square, rearrange the equation's terms to isolate and transform the quadratic part into a square.
• This conversion provides insights into the figure's geometric interpretation on the complex plane.

In our exercise, completing the square allowed us to reframe the equation \(x^2 - x + \frac{1}{4} + y^2 + 2iy\) into a circle equation. This enabled us to examine the nature of the path that the complex number \(z\) conforms to, illustrating its passage through the origin.