Problem 245 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(A\) be the A.M. and \(H\) the H.M. between two numbers \(a\) and \(b\) then \(\frac{a-A}{a-H} \times \frac{b-A}{b-H}=\frac{A}{H}\).

Step-by-Step Solution

Answer

By calculating the A.M. and H.M. of two numbers \(a\) and \(b\) and then substituting these values into the given formula, followed by algebraic simplification, we can prove the equation \(\frac{a-A}{a-H} \times \frac{b-A}{b-H}=\frac{A}{H}\), where A is the A.M. and H is the H.M. of the two numbers.

Step 1: Find A.M. and H.M.

Firstly, let's calculate the A.M. (A) and H.M. (H) of \(a\) and \(b\). A.M. \(A=(a+b)/2\) and H.M. \(H=2ab/(a+b)\).

Step 2: Substitute values of A and H

Now we substitute these values into the given formula: \(\frac{a-((a+b)/2)}{a-(2ab/(a+b))} \times \frac{b-((a+b)/2)}{b-(2ab/(a+b))}=\frac{(a+b)/2}{2ab/(a+b)}\). This step may require some algebraic manipulations and simplifications.

Step 3: Simplify the equation

Perform algebraic simplification to arrive at the required solution. If all steps have been executed correctly, both sides of the equation will equal each other, proving the initial equation.

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to make them easier to work with. In this exercise, it plays a key role. We need to simplify expressions involving the arithmetic mean (A.M.) and harmonic mean (H.M.). When given the A.M. as \(A = \frac{a+b}{2}\), and the H.M. as \(H = \frac{2ab}{a+b}\), we have to re-arrange and simplify the equation:

• Restate the terms in the fraction: \(\frac{a - A}{a - H}\) and \(\frac{b - A}{b - H}\).
• Simplify the arithmetic by finding common denominators or canceling terms.
• Ensure the expressions are simplified correctly to maintain balance in the equation.

Working through each step gives clearer understanding of the properties of numbers. This process not only shows the relationships between means but also strengthens problem-solving skills through algebraic reasoning.
Mastery comes from practice and exploring different ways to manipulate the equation until the required result is obtained.

Symmetric Functions

Symmetric functions are those whose values do not change if the variables are permuted. In this case, the concepts of A.M. and H.M. might seem different, but they beautifully demonstrate symmetry.- The definition of A.M. and H.M. shows a symmetric relationship between the numbers \(a\) and \(b\).- Regardless of the order of \(a\) and \(b\), both the means give us results with the same value because each mean considers the weighted effects of \(a\) and \(b\) equally.By observing that \(A = \frac{a+b}{2}\) and \(H = \frac{2ab}{a+b}\), you can see how they are constructed symmetrically by terms \(a+b\) and \(2ab\). Each mean is designed to handle numbers quantitatively differently, yet they express properties of symmetry in their basic formulation.
Understanding symmetric functions helps in recognizing these balanced relationships and simplifies the approach to working with algebraic expressions based on such properties.

Mean Inequality

The mean inequality typically involves relationships like AM-GM-HM inequality, where the arithmetic mean is always greater than or equal to the geometric mean, which in turn is always greater than or equal to the harmonic mean. In this exercise, - We observe how the A.M. and H.M. relate in terms of proving equal distribution or balance. - The equation provided stands as a test of how these means can equivalently balance each other within a set relationship. The challenge hints at understanding not just the calculation of means but also their inherent relationships. By mastering mean inequality, you learn:

• How different types of means compare to each other for given numbers.
• To apply these differences in finding proofs and solutions that rely on balanced relationships stemming from inequalities.

This exercise hones your ability to apply inequalities in solving algebraic problems, developing deeper insights into the logical flow and the implications of mean values.