Problem 18 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a, b, c\) are distinct positive real numbers and \(a^{2}+b^{2}\) \(+c^{2}=1\), then \(a b+b c+c a\) is (A) less than 1 (B) equal to 1 (C) greater than 1 (D) any real number

Step-by-Step Solution

Answer

The expression \(ab + bc + ca\) is less than 1 (Option A).

Step 1: Understanding the problem

We are given three distinct positive real numbers \(a, b,\) and \(c\) such that the sum of their squares is equal to 1. We need to find the relation of the sum of their pairwise products, \(ab + bc + ca\), relative to 1.

Step 2: Applying the inequality

Given \(a^2 + b^2 + c^2 = 1\), recall that for any real numbers, the inequality \(a^2 + b^2 + c^2 \geq ab + bc + ca\) holds. This is derived from the fact that \((a-b)^2 + (b-c)^2 + (c-a)^2 \geq 0\).

Step 3: Analyzing the inequality

From the equality given \(a^2 + b^2 + c^2 = 1\) and the inequality \(a^2 + b^2 + c^2 \geq ab + bc + ca\), it follows that \(1 \geq ab + bc + ca\).

Step 4: Checking distinctness condition

Since \(a, b, c\) are distinct, the equality \(a^2 + b^2 + c^2 = ab + bc + ca\) cannot hold, leading to \(ab + bc + ca < 1\). Distinctness ensures that none of the sums of squares equals the square, precluding equality in condition.

Key Concepts

Algebraic Identities

Algebraic identities are fundamental tools in mathematics that help us simplify and manipulate expressions. An identity is a statement that is true for all values of the variables within a certain set. In the context of our exercise, the identity for the sum of squares and the sum of pairwise products is crucial.We know that for any real numbers, the inequality \(a^2 + b^2 + c^2 \geq ab + bc + ca\) holds. This algebraic identity allows us to compare expressions in different forms:

• \((a - b)^2 + (b - c)^2 + (c - a)^2 \geq 0\) is always true because squares are non-negative.
• The expansion of this identity leads to the rearrangement inequality, which helps establish relationships between sums of squares and products.

Understanding this identity allows us to derive further conclusions about the relative magnitudes of different algebraic expressions. In essence, algebraic identities like this are powerful because they provide a general truth that simplifies complex mathematical problems.

Real Numbers

Real numbers are numbers that can be found on the number line. This includes all the integers, fractions, and numbers with decimals. In our exercise, we consider three distinct positive real numbers \(a, b,\) and \(c\). These numbers are crucial because:

• They are positive, meaning \(a, b, c > 0\).
• They are distinct, which means \(a eq b eq c\).

The importance of distinct positive real numbers affects the solution since it rules out any scenario where equality holds in the identity \(a^2 + b^2 + c^2 = ab + bc + ca\). Real numbers give us the flexibility to represent a wide range of quantities and facilitate the usage of inequalities, which is key in algebraic manipulations.Real numbers encompass an infinite set of values, making them a robust choice for creating problem scenarios in mathematics. They allow us to explore various possibilities and help in applying theoretical identities like the one used in this exercise.

Mathematical Reasoning

Mathematical reasoning is the backbone of problem-solving in mathematics. It involves identifying patterns, making conjectures, and arriving at valid conclusions.Let's look at our exercise where we deduced that \(ab + bc + ca < 1\):

• Given that \(a^2 + b^2 + c^2 =1\), and using the inequality \(a^2 + b^2 + c^2 \geq ab + bc + ca\), we know \(1 \geq ab + bc + ca\).
• Since \(a, b,\) and \(c\) are distinct, the equality cannot hold, so it must be that \(ab + bc + ca < 1\).

Understanding the role of inequalities and distinctness allows us to leverage mathematical reasoning effectively. Distinct positive elements lead to strict inequalities under certain conditions, helping us conclude that (A) \(< 1\) is correct.Through mathematical reasoning, we not only solve problems but also gain insights into why certain statements or identities hold true. This reasoning is essential for students learning to navigate the complexities of mathematics and enhances their problem-solving skills.