Problem 95 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the roots of \((z-1)^{25}=2 \omega^{2}(z+1)^{25}\) (where \(\omega\) is a complex cube root of unity) are plotted in the argand plane, they lie on (A) a straight line (B) a circle (C) an ellipse (D) None of these

Step-by-Step Solution

Answer

The roots lie on a circle in the argand plane.

Step 1: Identify Expression for Roots

We start by expressing the equation \((z-1)^{25} = 2 \omega^{2} (z+1)^{25}\) in terms of roots. To simplify, divide both sides by \((z+1)^{25}\), yielding:\[\left( \frac{z-1}{z+1} \right)^{25} = 2 \omega^{2}.\]Given that \(\omega\) is a complex cube root of unity, \(\omega^3 = 1\).

Step 2: Understanding the Implications of Cube Root of Unity

The cube roots of unity are \(1, \omega, \omega^2\), and \(\omega^2 = \overline{\omega}\). Hence \(2 \omega^2\) is a complex constant. This implies \(\frac{z-1}{z+1} = k\omega^{n/25}\), where \(k = \sqrt[25]{2},\) and \(n\) is an integer from 0 to 24 representing each of the 25 roots.

Step 3: Transform the Equation

Let \(w = \frac{z-1}{z+1}\). Then the equation \(w^{25} = 2 \omega^2\) implies that \(w\) is on a circle centered at the origin on the complex plane with radius \(\sqrt[25]{2}\).

Step 4: Express z in terms of w

Solve for \(z\) from \(w = \frac{z-1}{z+1}\), giving:\[z = \frac{1+w}{1-w}.\]This is a transformation known to map circles and lines in the \(w\)-plane to circles or lines in the \(z\)-plane.

Step 5: Determine Geometric Shape in z-plane

The set of points \(w\) that form a circle centered at the origin is mapped to a circle in the \(z\)-plane using the Möbius transformation \(z = \frac{1+w}{1-w}\). Therefore, the 25 roots form a circle in the complex plane.

Key Concepts

Argand Plane

The Argand Plane is a way to visualize complex numbers on a two-dimensional graph.
Each complex number corresponds to a point on this plane.

• The horizontal axis represents the real part of the number, denoted as \(\text{Re}(z)\).
• The vertical axis represents the imaginary part, denoted as \(\text{Im}(z)\).

Each complex number \(z = a + bi\) is plotted as the point \((a, b)\) on the plane.
This representation allows us to visualize operations on complex numbers, like addition or multiplication, geometrically.For instance, if complex roots are plotted on the Argand Plane, you can easily see properties such as symmetry or periodicity.
In the context of the given equation, plotting the roots helps in identifying their geometric arrangement, which in this case turns out to be a circle.

Roots of Unity

Roots of unity are special numbers in complex analysis that are solutions of the equation \(z^n = 1\).

• For cube roots of unity, \(n = 3\), and the roots are \(1, \omega, \omega^2\).
• These roots can be plotted on the Argand Plane, where they form an equilateral triangle with the origin as the center.

The fundamental property of these roots is that they evenly divide the unit circle in the complex plane.
This symmetry can be incredibly useful in solving polynomial equations as seen in the given exercise.
Here, \(\omega^2\) plays a significant role as a multiplier that affects how the roots are distributed on the Argand Plane.

Complex Roots

Complex roots are solutions to polynomial equations involving complex numbers, like the one in the exercise. These roots can be real or purely imaginary.
Often, they come in conjugate pairs, meaning if \(a + bi\) is a root, \(a - bi\) is also a root.In the context of plotting these roots on the Argand Plane, one can easily see patterns that might not be evident in a purely numeric format.
The pattern or shape that these roots form can often tell you more about the nature of the equation itself.In the given exercise, determining the geometric shape formed by the roots involved considering the transformation of the roots and mapping them from the \(w\)-plane to the \(z\)-plane, confirming their arrangement as a circle.

Möbius Transformation

A Möbius transformation is a function that takes a complex number \(w\) and maps it to another complex number \(z\) using a specific formula.
The general form is \(z = \frac{aw + b}{cw + d}\), where \(a, b, c,\) and \(d\) are complex numbers with \(ad - bc eq 0\).Möbius transformations are interesting because they map circles and lines in the \(w\)-plane to circles or lines in the \(z\)-plane.For example, in the exercise solution, we encounter the transformation \(z = \frac{1+w}{1-w}\).
This specific transformation allows for the visualization of how the roots forming a circle in the \(w\)-plane would form a circle in the \(z\)-plane too.The Möbius transformation is a powerful tool in complex analysis, often simplifying the understanding of geometric structures in the complex plane.