Problem 120 - A Complete Resource Book in Mathematics for JEE Main

Question

The three vertices of a triangle are represented by the complex numbers \(0, z_{1}\) and \(z_{2}\). If the triangle is equilateral, then (A) \(z_{1}^{2}+z_{2}^{2}+z_{1} z_{2}=0\) (B) \(z_{1}^{2}+z_{2}^{2}=z_{1} z_{2}\) (C) \(z_{2}^{2}-z_{1}^{2}=z_{1} z_{2}\) (D) \(z_{1}^{2}-z_{2}^{2}=z_{1} z_{2}\)

Step-by-Step Solution

Answer

The correct answer is (C): \(z_{2}^{2}-z_{1}^{2}=z_{1} z_{2}\).

Step 1: Understand the Properties of an Equilateral Triangle

In an equilateral triangle, all sides are equal, and the angles are each 60 degrees. For complex numbers representing vertices, this means that multiplying by \(e^{i \frac{2\pi}{3}}\) will rotate a point by 60 degrees.

Step 2: Define the Relationship Between Vertices

Since one vertex is at the origin, the other two points, represented by \(z_{1}\) and \(z_{2}\), must be such that rotating \(z_{1}\) by 60 degrees about the origin results in \(z_{2}\). Thus, \(z_{2} = z_{1} \cdot e^{i \frac{2\pi}{3}}\). Similarly, rotating \(z_{2}\) by 60 degrees should give \(z_{1}\).

Step 3: Set Up the Equation Based on Rotation

Since \(e^{i \frac{2\pi}{3}} = -\frac{1}{2} + i\frac{\sqrt{3}}{2}\), when you multiply \(z_{1}\) by this number, you should get \(z_{2}\). Therefore, \(z_{2} = z_{1}(-\frac{1}{2} + i\frac{\sqrt{3}}{2})\).

Step 4: Simplify the Expression

Rearrange the equation from Step 3 to form a relationship involving \(z_{1}^2\) and \(z_{1}z_{2}\), using the identity that a valid rotation in the complex plane means: \(z_{2}^2 - z_{1}^2 = z_{1}z_{2}\).

Step 5: Verify with Given Options

From the steps above, we've derived that: \(z_{2}^2 - z_{1}^2 = z_{1}z_{2}\). This matches option (C) exactly, confirming its validity.

Key Concepts

Equilateral Triangle

An equilateral triangle is a unique geometric figure where all three sides are equal in length, and all three interior angles are exactly 60 degrees. This balance of sides and angles gives it exceptional symmetry. In the context of complex numbers, this idea translates to specific relationships between points on a complex plane.
When considering vertices of a triangle as complex numbers, say one vertex at the origin, and the other two are represented by complex numbers \(z_1\) and \(z_2\), equal sides translate to maintaining equal distance between each of these points.
This condition means that rotating any vertex by an angle of 60 degrees should yield the position of the next vertex, hence the property of equidistance is preserved through rotation. This characteristic is crucial in analyzing triangles algebraically using complex numbers.

Complex Plane Rotation

In the complex plane, rotation of a complex number is an elegant operation made possible through multiplication by specific complex numbers. If you want to rotate a point \(z\) by 60 degrees counterclockwise, you multiply \(z\) by \(e^{i \frac{\pi}{3}}\). For a rotation of 120 degrees, the multiplier is \(e^{i \frac{2\pi}{3}}\).
This is useful in forming equilateral triangles in the complex plane as seen with:

• The vertex \(z_2\) being the result of rotating \(z_1\) by 60 degrees; mathematically expressed as \(z_2 = z_1 \cdot e^{i \frac{2\pi}{3}}\).
• This expresses the perfect symmetry of equilateral triangles because further rotating \(z_2\) by another 60 degrees should give back the original rotated point.

Multiplying by these exponential forms of complex numbers leads to new coordinates, shifting the angle of the original complex number around the origin, thereby efficiently achieving a rotation.

Complex Number Multiplication

Multiplying complex numbers is a process similar to working with algebraic expressions, but it encapsulates geometric transformations such as rotations and scalings in the complex plane. When you multiply two complex numbers, their magnitudes are multiplied while their angles (represented in polar form) are added.
For example, if you have a complex number \(z_1\) and you multiply it by \(e^{i \frac{2\pi}{3}}\), the effect is:

• The magnitude of \(z_1\) is preserved.
• The angle of \(z_1\) is increased by 120 degrees.

This is precisely why multiplying \(z_1\) by \(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\)—which is the equivalent of \(e^{i \frac{2\pi}{3}}\)—results in a complex number that represents \(z_2\)'s position, assuming an equilateral triangle's vertex arrangement.
This multiplication, therefore, encodes both rotation and the specific geometric configuration needed to establish equidistant spacing between the triangle's vertices on the plane.