Problem 162 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(z_{1}, z_{2}, z_{3}\) are the vertices of an equilateral triangle in the complex plane and \(z_{0}\) is the centroid, then (A) \(\frac{1}{z_{1}-z_{2}}+\frac{1}{z_{2}-z_{3}}+\frac{1}{z_{3}-z_{1}}=0\) (B) \(\left(z_{1}-z_{2}\right)^{2}+\left(z_{2}-z_{3}\right)^{2}+\left(z_{3}-z_{1}\right)^{2}=0\) (C) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=3 z_{0}^{2}\) (D) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}\)

Step-by-Step Solution

Answer

Answers are (A), (B), and (D).

Step 1: Understanding the Equilateral Triangle Formula

For an equilateral triangle in the complex plane with vertices \(z_1, z_2, z_3\), it follows a particular property: \(z_1 + z_2 + z_3 = 3z_0\), where \(z_0\) is the centroid of the triangle.

Step 2: Check Option A

Consider the expression \(\frac{1}{z_1 - z_2} + \frac{1}{z_2 - z_3} + \frac{1}{z_3 - z_1}\). This expression sums to zero for an equilateral triangle as all the sides are equal and symmetric around the centroid. Therefore, \(\boxed{(A)\text{ is correct.}}\).

Step 3: Analyze Option B

For an equilateral triangle, each distance is the same, thus the squared differences \((z_1 - z_2)^2 + (z_2 - z_3)^2 + (z_3 - z_1)^2\) can simplify to zero, following the principle of symmetry and balance in complex form for equilateral figures. Thus, \(\boxed{(B)\text{ is correct.}}\).

Step 4: Evaluate Option C

Using the property \(z_1 + z_2 + z_3 = 3z_0\), square both sides: \((z_1 + z_2 + z_3)^2 = 9z_0^2\). This expands to \(z_1^2 + z_2^2 + z_3^2 + 2(z_1z_2 + z_2z_3 + z_3z_1) = 9z_0^2\), which doesn't balance with \(z_1^2 + z_2^2 + z_3^2 = 3z_0^2\) unless terms are rearranged incorrectly in a more specific context. This is false under general calculations, so \(\boxed{(C)\text{ is incorrect.}}\).

Step 5: Verify Option D

Given \(z_1 + z_2 + z_3 = 3z_0\) and squared form earlier addressed, if you expand the formed expressions \(z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1\), the relationship provided is particular to the equilateral triangle, satisfying the condition on balancing. So \(\boxed{(D)\text{ is correct.}}\).

Key Concepts

Equilateral Triangle

An equilateral triangle is a geometric figure where all three sides are of equal length. In the context of complex numbers, the vertices of such a triangle can be represented as complex numbers. This representation can simplify understanding and working with geometric properties using algebraic techniques. When dealing with complex numbers, each vertex of the triangle can be considered as a point on the complex plane, with the complex plane representing the coordinate system where real numbers are plotted on the x-axis and imaginary numbers on the y-axis.

• Each side of the triangle is equal, hence any rotation or reflection will not change its form.
• They are also equiangular, meaning each angle inside the triangle is 60 degrees.
• In terms of complex arithmetic, rotations and reflections are straightforward to compute using multiplication by complex numbers, lending well to symmetry.

Equilateral triangles are very symmetrical and thus have unique properties, such as all medians, altitudes, and internal angle bisectors coincide at the same point, known as the centroid. This makes them particularly fascinating when combined with complex analysis.

Complex Plane

The complex plane is a two-dimensional plane used to graph complex numbers. Each complex number is represented as a point where the horizontal axis represents the real part and the vertical axis represents the imaginary part. It provides a visual approach for understanding complex numbers and their operations like addition, subtraction, and multiplication.

• On this plane, a complex number such as \( z = a + bi \) is represented as the point \((a, b)\).
• The modulus of a complex number denotes its distance from the origin \((0,0)\) and can be calculated as \( |z| = \sqrt{a^2 + b^2} \).
• The argument of a complex number is the angle (usually measured in radians) from the positive real axis to the line that represents the complex number.

Using the complex plane to discuss the properties of geometric shapes, such as triangles, brings spatial intuition into algebraic form, making it easier to analyze symmetries and transformations.

Centroid of Triangle

The centroid of a triangle is the point where all three medians intersect. It is often referred to as the triangle's "center of gravity" and is the average position of all the points in the shape. In the context of complex numbers and geometry, the centroid remains crucial.

• For vertices \(z_1, z_2, z_3\) in the complex plane, the centroid \(z_0\) is calculated as \(z_0 = \frac{z_1 + z_2 + z_3}{3}\).
• The centroid divides each median into a 2:1 ratio, with the centroid being twice as close to the vertex as it is to the midpoint of the opposite side.
• In equilateral triangles, due to their symmetry, the centroid is equidistant from all three vertices.

Understanding the centroid helps in exploring and confirming properties of triangles, especially equilateral triangles, providing insight into balance and symmetry.