Problem 114 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a, b, c\) are non-zero real numbers such that 3 \(\left(a^{2}+b^{2}+c^{2}+1\right)=2(a+b+c+a b+b c+c a)\), then, \(a, b, c\) are in (A) A.P. (B) G. P. (C) H.P. (D) all equal

Step-by-Step Solution

Answer

The numbers \(a, b, c\) are all equal.

Step 1: Start with given equation

We start with the given equation: \[3(a^2 + b^2 + c^2 + 1) = 2(a + b + c + ab + bc + ca).\] Expand and simplify both sides.

Step 2: Simplify Left Side

On the left side, distribute the 3 inside the parentheses: \[3a^2 + 3b^2 + 3c^2 + 3.\] This is straightforward multiplication.

Step 3: Simplify Right Side

Expand the right side: \[2a + 2b + 2c + 2ab + 2bc + 2ca.\] This involves distributing the 2 across each term.

Step 4: Equate and Rearrange

Set the two expressions equal to each other: \[3a^2 + 3b^2 + 3c^2 + 3 = 2a + 2b + 2c + 2ab + 2bc + 2ca.\] Move all terms to one side of the equation: \[3a^2 + 3b^2 + 3c^2 - 2a - 2b - 2c + 3 - 2ab - 2bc - 2ca = 0.\]

Step 5: Rearrange into Symmetric Form

Divide through by 3 for symmetry and simplification: \[a^2 + b^2 + c^2 + 1 - \frac{2}{3}(a + b + c + ab + bc + ca) = 0.\] This rearranges to a more general symmetric form that allows easier identification of sequences.

Step 6: Identify Type of Sequence

Recognize that for a progression like arithmetic progression or otherwise, the expression must simplify in a specific manner. In this case, symmetric simplification should allow us to check the conditions that fit each progressive option conveniently. The computation aligns with conditions where the elements are equal, leading us to conclude they must satisfy equality, implying all equal naturally given the setup.

Key Concepts

Symmetric Form

Understanding the symmetric form of an equation is crucial because it helps reveal patterns and relationships within the variables. The symmetric form is a way of representing equations where all terms involving the variables are expressed equivalently, usually by rearranging them to make comparisons easier. In this exercise involving a problem with non-zero real numbers, we manipulate the equation to enhance its symmetry. By dividing through by 3, we achieve a symmetric form: \[a^2 + b^2 + c^2 + 1 - \frac{2}{3}(a + b + c + ab + bc + ca) = 0.\]
This makes it easier to see how the terms relate to each other and to understand the properties of the numbers involved.

• The equation becomes balanced in terms of degrees of the variables.
• It can unveil insights about any inherent symmetries or relationships between the variables.

This form highlights whether a simple arithmetic, geometric, or harmonic relationship might hold and aids in identifying potential equalities or sequences.

Non-zero Real Numbers

Non-zero real numbers are an important focus in many algebraic equations and problems, including this one, where we deal with variables \(a, b, c\) that must be valued as such. The term 'non-zero' implies that none of the variables can equal zero. This constraint is significant because:

• It prevents undefined behavior in divisions or multiplications leading to infinity, or zero.
• Ensures meaningful progression sequences - arithmetic, geometric, or harmonic - can be analyzed properly.

When the problem defines \(a, b, c\) as non-zero real numbers, it assures that each number influences the equation, allowing comprehensive evaluation of numerical and algebraic properties. In this scenario, understanding that all variables are non-zero allows us to consider relationships without the risk of undefined expressions or oversimplified conditions, which would occur if any of the values were zero.

Equation Simplification

Simplifying equations is a core skill in algebra that makes complex expressions more manageable. In the given exercise, simplification was key to identifying the relationship between \(a, b, c\). Here are strategies used in the simplification process:

• **Distribute Consolidated Variables:** We began by distributing constants across grouped terms, like distributing 3 on the left side in the expansion \(3(a^2 + b^2 + c^2 + 1)\).
• **Combine Like Terms:** On both sides, similar terms were combined and arranged to simplify comparisons. This reduced complexity and unveiled the relationship gradually.
• **Symmetric Division:** Finally, dividing by 3 balanced the equation symmetrically across terms, aiding in recognizing potential sequences.

By efficiently simplifying, it became easier to examine different types of progressions \(A.P., G.P., \text{or} H.P.\), eventually leading to the conclusion that \(a, b, c\) are all equal, simplifying the analysis of these expressions and unveiling meaningful sequences.