Problem 153 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(z_{1}, z_{2}, z_{3}\) are three points lying on the circle \(|z|=2\) then the minimum value of \(\left|z_{1}+z_{2}\right|^{2}+\left|z_{2}+z_{3}\right|^{2}+\) \(\left|z_{3}+z_{1}\right|^{2}\) is equal to (A) 6 (B) 12 (C) 15 (D) 24

Step-by-Step Solution

Answer

The minimum value is 24.

Step 1: Understand the Geometry

The given points \(z_1, z_2, z_3\) lie on the circle \(|z| = 2\), which means that each point \(z_i\) satisfies the equation \(|z_i| = 2\). This is a circle centered at the origin with radius 2 in the complex plane.

Step 2: Use the Property of Circle

Since \(z_1, z_2, z_3\) are on the circle \(|z|=2\), their magnitudes are the same. We need to find the minimum value of the expression: \[\left|z_1 + z_2\right|^2 + \left|z_2 + z_3\right|^2 + \left|z_3 + z_1\right|^2.\]

Step 3: Expand Each Term

Begin by expanding the expression, making use of the property \(|a + b|^2 = |a|^2 + |b|^2 + 2\Re\{a\overline{b}\}\):- \(|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + 2\Re\{z_1\overline{z_2}\}\)- \(|z_2 + z_3|^2 = |z_2|^2 + |z_3|^2 + 2\Re\{z_2\overline{z_3}\}\)- \(|z_3 + z_1|^2 = |z_3|^2 + |z_1|^2 + 2\Re\{z_3\overline{z_1}\}\)

Key Concepts

Geometry of Complex Numbers

Complex numbers provide a fascinating bridge between algebra and geometry. This dual nature allows us to visualize and interpret complex operations in a geometric manner. In the complex plane, each complex number is represented as a point. The horizontal axis represents the real part, while the vertical axis denotes the imaginary part.

When we work with complex numbers like our given points \(z_1, z_2, z_3\), these points are plotted on the complex plane. Understanding the geometry involves interpreting these points' locations and operations around them. For instance, when we add two complex numbers, we are essentially translating the points across the plane. Every operation has a geometric implication which can make understanding abstract concepts easier.

In the case of our exercise, we leverage the geometric property that all points \(z_1, z_2, z_3\) lie on a circle. This circle provides a clear boundary and helps simplify potential complex calculations.

Circle in the Complex Plane

A circle in the complex plane is defined by its center and radius, similar to circles in the Cartesian plane. In our problem, the circle is centered at the origin \((0,0)\) with a radius of 2. Hence, any complex number \(z\) satisfies the equation \(|z| = 2\).

• This means that any point \(z_i\) lying on this circle has a magnitude (distance from the origin) equal to 2.
• The circle implies that all points \(z_1, z_2, z_3\) have an equal importance, and any transformation needs to take their location on the circle into account.

Because these points are constrained to this circular structure, they offer symmetry which can simplify calculations like the minimum of the given expression. The geometric insight here is crucial as it limits the range and possibility of values that can be taken by the properties of the numbers, allowing for simplifications from complex algebraic expressions.

Properties of Magnitude

Magnitude in the context of complex numbers refers to the distance a number is from the origin in the complex plane. It is calculated using the formula \(|z| = \sqrt{a^2 + b^2}\), where \(z = a + bi\). For our problem, the magnitude of all points \(z_1, z_2, z_3\) is given as 2.

• Understanding magnitude is crucial since it remains unchanged under rotation. Hence, any two complex numbers that have the same magnitude lie on a circle with a constant radius.
• In this exercise, the magnitude properties help us know that the sums \(|z_1 + z_2|\), \(|z_2 + z_3|\), and \(|z_3 + z_1|\) are also confined by similar magnitude rules, allowing calculations using symmetry and geometry.
• Expansion using \(|a + b|^2 = |a|^2 + |b|^2 + 2\Re\{a\overline{b}\}\) allows decomposing complex expressions by calculating magnitudes directly.

The properties of magnitude simplify understanding of distances and interactions within the complex plane, providing a foundational concept necessary for more advanced calculations.