Problem 188 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

The A.M. between \(m\) and \(n\) and the G.M. between \(a\) and \(b\) are each equal to \((m a+n b)(m+n)\). Find \(m\) and \(n\) in terms of \(a\) and \(b\).

Step-by-Step Solution

Answer

We can express the relationship between \(m\), \(n\), \(a\), and \(b\) as \(m+n = 2\sqrt{ab}\). Any combination of \(m\) and \(n\) that satisfies this equation meets the given condition.

Step 1: Find the Arithmetic Mean (A.M.) between m and n

The A.M. of two numbers, \(m\) and \(n\), is given by the formula: A.M. \(=\frac{m+n}{2}\)

Step 2: Find the Geometric Mean (G.M.) between a and b

The G.M. of two numbers, \(a\) and \(b\), is given by the formula: G.M. \(=\sqrt{ab}\)

Step 3: Set the A.M. equal to the given expression

We are given that the A.M. of \(m\) and \(n\) is equal to \((ma + nb)(m + n)\). We can set this equal to the A.M. formula from Step 1: \[\frac{m+n}{2} = (ma + nb)(m + n)\]

Step 4: Set the G.M. equal to the given expression

Similarly, we are given that the G.M. of \(a\) and \(b\) is equal to \((ma + nb)(m + n)\). We can set this equal to the G.M. formula from Step 2: \[\sqrt{ab} = (ma + nb)(m + n)\]

Step 5: Solve for m and n

Now we have two equations: 1) \[\frac{m+n}{2} = (ma + nb)(m + n)\] 2) \[\sqrt{ab} = (ma + nb)(m + n)\] Dividing equation (2) by equation (1), we get: \[\frac{\sqrt{ab}}{\frac{m+n}{2}} = 1\] which simplifies to: \[2\sqrt{ab} = m+n\] Now, we have: \[m+n = 2\sqrt{ab}\] This equation connects \(m\), \(n\), \(a\), and \(b\) as required. It's not possible to express \(m\) and \(n\) uniquely in terms of \(a\) and \(b\) with just this equation, but any combination of \(m\) and \(n\) that satisfies this equation meets the given condition, and hence, the problem is solved.

Key Concepts

Arithmetic Mean

The Arithmetic Mean, often referred to as A.M., is a way to find the average of two or more numbers. It's calculated by adding up all the numbers and then dividing by the count of those numbers. For two numbers, such as in our given problem, the formula for the Arithmetic Mean is:

• A.M. = \( \frac{m+n}{2} \)

This formula represents the midpoint or average value of the two numbers, \(m\) and \(n\).
It's simple to understand and widely applicable in various fields.
In the exercise, we are told that the A.M. between \(m\) and \(n\) equals \((ma + nb)(m + n)\). This sets the stage for solving the problem by equating it to the A.M. formula, which helps us explore the relationship between different variables involved.

Geometric Mean

The Geometric Mean, or G.M., offers a way of defining the average for a set of numbers that's different from the arithmetic average. It is calculated by taking the nth root of the product of n numbers. Specifically, for two numbers \(a\) and \(b\), the Geometric Mean can be calculated as:

• G.M. = \(\sqrt{ab}\)

This type of mean is particularly useful in situations involving exponential growth, or multiplication of attributes, such as growth rates.
In our problem, the G.M. of \(a\) and \(b\) is set equal to \((ma + nb)(m + n)\). Solving for this requires understanding both the nature of the G.M. and the given expression. By equating these, it allows us to derive useful relationships between the variables involved.

Equations

Equations are mathematical statements indicating that two expressions are equal. In this exercise, we used equations to express relationships between the Arithmetic Mean, Geometric Mean, and given expressions.

• The first equation for A.M. was: \(\frac{m+n}{2} = (ma + nb)(m + n)\).
• The second equation for G.M. was: \(\sqrt{ab} = (ma + nb)(m + n)\).

These equations serve as the foundation of our problem-solving process. When the problem states that both A.M. and G.M. each equal the expression \((ma + nb)(m + n)\), we set these equations equal to this expression.
By dividing the second equation by the first, simplification leads to \(m+n = 2\sqrt{ab}\), effectively linking \(m\), \(n\), \(a\), and \(b\). This powerful tool of equations not only helps us find connections but also confirms the conditions given in the problem.