Problem 89 - 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

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

Step 1: Understand the Problem

We need to determine the geometric relationship between the complex numbers \(z_1\), \(z_2\), and the origin given the equation \(\frac{z_1}{z_2}+\frac{z_2}{z_1}=1\).

Step 2: Simplify the Equation

Start by multiplying both sides by \(z_1 z_2\) to eliminate the fractions:\[\frac{z_1}{z_2} + \frac{z_2}{z_1} = 1 \Rightarrow z_1^2 + z_2^2 = z_1 z_2.\]

Step 3: Analyze the Equation

Reorganize the simplified equation: \[z_1^2 - z_1 z_2 + z_2^2 = 0.\] This resembles the form of the cosine rule for a triangle.

Step 4: Identify Geometric Implications

The equation \(z_1^2 - z_1 z_2 + z_2^2 = 0\) can be interpreted as a condition for roots of unity or an equilateral triangle. This implies that \(z_1\), \(z_2\), and the origin form an equilateral triangle.

Step 5: Conclusion

With the derived equation resembling that of the roots of unity, \(z_1\) and \(z_2\), and the origin form an equilateral triangle. So the correct choice is (C).

Key Concepts

Geometric Interpretation

To understand the nature of complex numbers, it is often helpful to visualize them as points on a plane. This is the crux of their geometric interpretation. Each complex number, such as \( z = a + bi \), corresponds to a point \((a, b)\) in the complex plane:

• The real part \(a\) corresponds to the horizontal axis.
• The imaginary part \(b\) corresponds to the vertical axis.

In this plane, complex numbers can also be represented as vectors. The length of this vector is the magnitude of the complex number, calculated using the formula \( |z| = \sqrt{a^2 + b^2} \).

When complex numbers are part of an equation, such as \( \frac{z_1}{z_2} + \frac{z_2}{z_1} = 1 \), their positions relative to each other and the origin often have direct implications for their geometric arrangement. By solving the equation, we gain insights into this arrangement, which in our example, suggests a symmetrical shape like an equilateral triangle.

Equilateral Triangle

The solution of the equation \( z_1^2 - z_1 z_2 + z_2^2 = 0 \) points to the formation of an equilateral triangle between the complex numbers and the origin. An equilateral triangle in the complex plane is characterized by all sides being equal in length and all angles measuring \(60^\circ\).

To visualize it easily:

• Consider \( z_1 \), \( z_2 \), and the origin as vertices of the triangle.
• The equal magnitude of the segments formed by \( z_1 - z_2 \), \( z_1 - 0 \), and \( z_2 - 0 \) confirms the equal sides property.
• For these vectors forming such an angle, the scalar product will align with cosine rule derivations found in equal energetic setups.

This symmetry around the origin is an important part of understanding the geometric relations in the complex plane, also relating closely with the concept of the roots of unity simplifying this geometric interpretation.

Roots of Unity

Roots of unity are a fundamental concept when it comes to complex numbers and their symmetric arrangements. Simply put, the \( n \)-th roots of unity are complex numbers which satisfy the equation \( z^n = 1 \).

They are evenly distributed on the unit circle, which is a circle with radius 1 centered at the origin of the complex plane. For case \( n=3 \) (like forming an equilateral triangle with origin):

• The 3rd roots of unity are \( 1, \omega, \omega^2 \) where \( \omega = e^{i \frac{2\pi}{3}} \).
• These points form vertices of an equilateral triangle on the plane.
• The symmetry and equal spacing attribute to equilateral triangle formation when seen from the origin.

Understanding roots of unity helps illustrate why an equation like \( z_1^2 - z_1 z_2 + z_2^2 = 0 \) could align the complex numbers like an equilateral triangle, as both involve principles of harmonious distribution and balance in the complex plane.