Problem 117 - A Complete Resource Book in Mathematics for JEE Main

Question

If the origin and the two points represented by complex numbers \(A\) and \(B\) form vertices of an equilateral triangle, then \(\frac{A}{B}+\frac{B}{A}=\) (A) 1 (B) - I (C) 2 (D) None of these

Step-by-Step Solution

Answer

1

Step 1: Understanding the Problem

We need to determine the value of the expression \(\frac{A}{B} + \frac{B}{A}\) given that the origin, and the points represented by complex numbers \(A\) and \(B\), form an equilateral triangle.

Step 2: Equilateral Triangle Property

For points in the complex plane, if three points form an equilateral triangle, the triangle formed by their complex numbers is similar to the triangle in the same ratio of rotation.

Step 3: Using Rotation Property

When two complex numbers \(A\) and \(B\) form an equilateral triangle with the origin, then \(B = Ae^{i\frac{\pi}{3}}\) or \(B = Ae^{-i\frac{\pi}{3}}\) due to rotation by \(60^\circ\).

Step 4: Substitute Values into Expression

Insert \(B = Ae^{i\frac{\pi}{3}}\) into the expression: \(\frac{A}{B} + \frac{B}{A} = \frac{A}{Ae^{i\frac{\pi}{3}}} + \frac{Ae^{i\frac{\pi}{3}}}{A} = e^{-i\frac{\pi}{3}} + e^{i\frac{\pi}{3}}\).

Step 5: Simplify the Expression

Using the identity \(e^{i\theta} + e^{-i\theta} = 2\cos(\theta)\), we find: \(e^{-i\frac{\pi}{3}} + e^{i\frac{\pi}{3}} = 2\cos\left(\frac{\pi}{3}\right) = 2\times\frac{1}{2} = 1\).

Step 6: Conclusion

Therefore, the value of the expression \(\frac{A}{B} + \frac{B}{A}\) is 1. The correct answer is option (A), 1.

Key Concepts

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides are equal in length, and all three internal angles are 60 degrees. In the complex plane, this property is particularly interesting, because it introduces a consistent rotation and scaling of complex numbers.
This specific triangle is often used in mathematics because it has symmetrical properties that simplify complex problems.

• In terms of geometry, if you place one vertex at the origin (0, 0) in the complex plane, then the other two vertices must also form equal angles, creating a consistent pattern.
• In our exercise, the points represented by complex numbers A and B, along with the origin, form such a triangle.

By exploring this property on a geometric level using complex numbers, it allows one to see how numerical relationships translate into consistent patterns and symmetries.

Complex Plane

The complex plane, also known as the Argand plane, is a two-dimensional plane used to visually represent complex numbers. The horizontal axis represents the real part of complex numbers, while the vertical axis represents the imaginary part.

• Each complex number can be viewed as a point in this plane or, equivalently, as a vector from the origin to this point.
• This plane helps to better understand operations involving complex numbers, especially when it comes to addition, multiplication, and even geometric transformations.

In the original problem, complex numbers A and B are seen as points (or vectors) in this plane. By observing how these points relate to each other through geometric interpretations, we can solve problems concerning their magnitudes and directions effectively.

Polar Representation of Complex Numbers

Polar representation is one of two ways to represent complex numbers, the other being the rectangular form. In polar form, a complex number is expressed using a magnitude (or modulus) and an angle (or argument) relative to the positive direction of the real axis.

• The expression for a complex number in polar form is: \[ z = r \cdot e^{i\theta} \] where \( r \) is the magnitude, and \( \theta \) is the angle.
• This form is particularly useful when performing multiplication or division of complex numbers.

In this problem, recognizing that rotation about the origin by 60 degrees or \( \frac{\pi}{3} \) radians can be represented as multiplying by \[ e^{i\frac{\pi}{3}} \] or \[ e^{-i\frac{\pi}{3}} \]. This highlights the beauty of using polar forms for transformation purposes, providing an elegant solution to geometric problems in the complex plane.