Problem 37 - A Complete Resource Book in Mathematics for JEE Main 2017

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

Option (C): \(z_1, z_2\), and the origin form an equilateral triangle.

Step 1: Introduce the Complex Equation

We begin with the given equation: \(\frac{z_1}{z_2} + \frac{z_2}{z_1} = 1\). Our goal is to understand the properties or relationship between these complex numbers \(z_1\) and \(z_2\).

Step 2: Simplify and Transform the Equation

Multiply both sides of the equation by \(z_1 z_2\) to eliminate the fractions, giving us: \(z_1^2 + z_2^2 = z_1 z_2\). This will be our working equation.

Step 3: Identify Geometric Properties

Notice that the equation \(z_1^2 + z_2^2 = z_1 z_2\) resembles a condition often associated with special triangles in geometric interpretations of complex numbers. Specifically, multiplying both sides by 4, we get \(4z_1^2 + 4z_2^2 = 4z_1 z_2\), which rearranges to \((2z_1)^2 + (2z_2)^2 = (2z_1)(2z_2)\). This is consistent with the properties of an equilateral triangle, where the sum of squares of any two sides is equal to the square of the third side.

Step 4: Analysis of Options

The derived equation indicates that \(z_1, z_2,\) and the origin trace out an equilateral triangle. Let's check against given options:- (A) Collinearity is not indicated.- (B) Right-angled triangle is not indicated.- (C) Indeed corresponds to an equilateral triangle.- (D) Option (C) fits the conclusion.

Key Concepts

Geometric Interpretation

The concept of geometric interpretation in complex numbers involves understanding how these numbers can represent points or vectors on a plane. Each complex number can be thought of as having two components: a real part and an imaginary part. Together, they define a point in a two-dimensional space that resembles a coordinate system.

When we deal with equations involving complex numbers like \(\frac{z_1}{z_2} + \frac{z_2}{z_1} = 1\), we are often looking at the relationships or geometrical arrangements formed by these points in the complex plane.

• A complex number \(z = a + bi\) represents a point \( (a, b) \) where \(a\) is the real part and \(b\) is the imaginary part.
• When you have multiple complex numbers, you can visualize lines, triangles, or even more complex shapes, depending on their particular relations.
• In our specific case, the equation transforms to highlight a triangular relation among the points defined by these complex numbers.

Understanding this geometric framework is essential to analyze how these points relate and form shapes like triangles.

Equilateral Triangle

An equilateral triangle is a special type of geometry where all three sides are of equal length, and all three angles are equal as well, each being 60 degrees. This property is pivotal in understanding the original exercise, as complex numbers can define such shapes.

In terms of complex numbers, imagine each number as a point or a vertex of a triangle. The equation \(z_1^2 + z_2^2 = z_1 z_2\) suggests that these points form an equilateral triangle when combined with the origin. Here's why:

• When we multiply both sides by 4, the equation rearranges to \((2z_1)^2 + (2z_2)^2 = (2z_1)(2z_2)\).
• This format matches the characteristics of an equilateral triangle in coordinate geometry—where the square of one side equals the sum of the squares of the others.
• Visualizing the complex numbers as vectors, their arrangement on a plane creates equal angles and distances with the origin.

Overall, viewing complex numbers as geometrical shapes on the plane unveils the nature of their relationships, highlighting the remarkable symmetry in an equilateral triangle formation.

Complex Equation

Complex equations often provide a bridge between algebra and geometry, revealing underlying relationships through algebraic manipulation. The exercise starts with a seemingly simple equation \(\frac{z_1}{z_2} + \frac{z_2}{z_1} = 1\) which, at first glance, may not appear straightforwardly intuitive regarding geometry.

However, by multiplying through by \(z_1 z_2\), we eliminate fractions, yielding \(z_1^2 + z_2^2 = z_1 z_2\). This transformation is key:

• By simplifying the equation, complex numbers are more tractably handled as sides or vectors in a geometric sense.
• It's then easier to identify symmetrical or specific geometric patterns, like triangles.
• Finally, algebraic techniques like multiplying both sides by the same factor help in visualizing and solving these equations geometrically.

Understanding how to manipulate these complex equations can reveal hidden geometrical insights, illuminating how numbers relate to constructs like equilateral triangles, making them an invaluable aspect of mathematical studies.