Problem 89 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(z_{1}\) and \(z_{2}\) be two complex numbers such that \(\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{1}}=1\), then (A) \(z_{1}, z_{2}\) are collinear (B) \(z_{1}, z_{2}\) and the origin from a right angled triangle (C) \(z_{1}, z_{2}\) and the origin form an equilateral triangle (D) None of these

Step-by-Step Solution

Answer

(C) \(z_{1}, z_{2}\) and the origin form an equilateral triangle.

Step 1: Consider the Given Equation

We are given the equation \(\frac{z_1}{z_2} + \frac{z_2}{z_1} = 1\). Our goal is to understand the relationship between \(z_1\) and \(z_2\).

Step 2: Multiply Through by \(z_1 z_2\)

Multiply the entire equation by \(z_1 z_2\) to eliminate the fractions: \(z_1^2 + z_2^2 = z_1 z_2\).

Step 3: Rearrange the Equation

Rearrange the equation to group similar terms: \(z_1^2 - z_1 z_2 + z_2^2 = 0\).

Step 4: Recognize a Factorable Quadratic Form

The equation \(z_1^2 - z_1 z_2 + z_2^2 = 0\) can be expressed in the quadratic form \((z_1 - \omega z_2)(z_1 - \omega^2 z_2) = 0\), where \(\omega\) is a primitive cube root of unity, satisfying \(\omega^3 = 1\) and \(\omega eq 1\).

Step 5: Solve for \(z_1\) in terms of \(z_2\)

This factorization implies \(z_1 = \omega z_2\) or \(z_1 = \omega^2 z_2\). This implies that \(z_1\) and \(z_2\) are related by a rotation of \(120^\circ\) or \(240^\circ\) in the complex plane, hence forming an equilateral triangle with the origin.

Step 6: Conclude Based on Given Options

Our analysis shows that \(z_1\) and \(z_2\) form an equilateral triangle with the origin since they are rotations of each other by \(120^\circ\) or \(240^\circ\). Therefore, the correct option is (C).

Key Concepts

Equilateral Triangle

In an equilateral triangle, all three sides are equal in length, and each angle measures 60 degrees. This concept extends beyond the real-number plane into the complex plane.
When dealing with complex numbers, an equilateral triangle can also consist of vertices that are complex numbers. Consider three points in the complex plane, which you will denote as complex numbers. If they form an equilateral triangle, the relationships between these points are special and distinct.
In this specific exercise, the relation between two complex numbers, \(z_1\) and \(z_2\) leads to the formation of an equilateral triangle with the origin. The origin acts alongside \(z_1\) and \(z_2\) as the triangle's vertices. Here, the equation \(\frac{z_1}{z_2} + \frac{z_2}{z_1} = 1\) simplifies to this geometric interpretation. When manipulated algebraically, these complex numbers are shown to represent rotations around these central points.

Roots of Unity

Roots of unity are solutions to the equation \(x^n = 1\), where \(n\) is a positive integer. For the cubic form \(x^3 = 1\), the roots are \(1\), \(\omega\), and \(\omega^2\), where \(\omega\) is a primitive cube root of unity. Primitive means that it is a generator of all other roots, excluding 1 itself.
These roots are represented in the complex plane as points that are equally spaced around the unit circle, forming vertices of an equilateral triangle. Each of these points represents a rotation by \(120^\circ\) around the origin.

• \(1\)
• \(\omega = e^{2\pi i / 3}\)
• \(\omega^2 = e^{4\pi i / 3}\)

These rotations, represented by \(\omega\) and \(\omega^2\), are critical for understanding the positioning of \(z_1\) and \(z_2\) in the complex plane as they are manipulated in the exercise to show how they form angles of \(120^\circ\) or \(240^\circ\).

Complex Plane

The complex plane is a two-dimensional plane used to represent complex numbers. It has a horizontal axis (real) and a vertical axis (imaginary).
Each complex number, \(z = a + bi\), can be plotted in this plane with \(a\) and \(b\) forming the coordinates. The unique aspect of the complex plane is its ability to visualize operations involving complex numbers as geometric transformations.
In the given problem, the complex plane helps to easily demonstrate rotations and transformations. For example, multiplying by a complex number located on the unit circle (like the roots of unity) results in a rotation. This is fundamental in showing how \(z_1\) rotates by \(120^\circ\) or \(240^\circ\) relative to \(z_2\), leading to their arrangement in an equilateral triangle. Thus, understanding the geometric properties of the complex plane allows deeper insight into the nature of these transformations and relationships.