Problem 251 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(2(y-a)\) is the H.M. between \(y-x\) and \(y-z\), then show that \(x-a, y-a, z-a\) are in G.P.

Step-by-Step Solution

Answer

The given condition implies that \(x-a, y-a, z-a\) are in geometric progression.

Step 1: Write the Harmonic Mean Formula

First, list down the Harmonic Mean formula. If \(a\) and \(b\) are two quantities, then the Harmonic Mean (H.M.) is given by \(H = 2ab / (a+b)\)

Step 2: Replace the H.M. by the given expression

Substitute the formula of H.M. into the given expression and simplify. So we get, \[2(y-a) = 2(y-x)(y-z) / [(y-x)+(y-z)]\]. Simplify it further to get [(y-x)+(y-z)](y-a) = 2(y-x)(y-z)

Step 3: Simplify

Now, simplify the equation further. The left hand side becomes \(2y-x-z = 2a\), and the right hand side becomes \(2yz - 2xz - 2yz + 2xz\). Hence the equation becomes, \(2y=x+z\)

Step 4: Prove the given numbers are in G.P.

For \(x-a, y-a, z-a\) to be in Geometric Progression (G.P.), it must satisfy the property that the ratio of any two successive terms should be constant. Therefore, \[(y-a)/(x-a) = (z-a)/(y-a)\]. Substituting \(2y=x+z\) in the above equation, we have \[\[(2y-a)/(x-a)\] = \[(2y-a)/(2y-x)\]. This simplifies to 1 = 1. Hence, the numbers \(x-a, y-a, z-a\) are in G.P.

Key Concepts

Geometric Progression

A geometric progression, or G.P., is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number known as the common ratio. In simpler terms, if you have the first term of the sequence as \(a\) and the common ratio as \(r\), the sequence looks like this:

• First term: \(a\)
• Second term: \(ar\)
• Third term: \(ar^2\)
• And so on...

In the exercise you worked on, you needed to prove that \(x-a, y-a, z-a\) follow a G.P. To verify this, the ratio between successive terms must be constant. Thus, \(\frac{y-a}{x-a} = \frac{z-a}{y-a}\) must hold true. After simplifying the problem using harmonic mean relationships, you find that indeed, these terms create a G.P.

Harmonic Progression

A harmonic progression (H.P.) is a sequence of numbers where the reciprocals of the terms form an arithmetic progression (A.P.). This means if you take the reciprocal of each term in an H.P., you'll form a sequence that has a common difference between terms.

• If the sequence is \(a, b, c, \ldots\), then their reciprocals \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \ldots\) form an A.P.
• In an H.P., you cannot sum the terms directly like in an A.P. Instead, you sum their reciprocals.

In the exercise, the quantity \(2(y-a)\) is the harmonic mean between \(y-x\) and \(y-z\). The harmonic mean formula \(H = \frac{2ab}{(a+b)}\) plays a vital role in deriving the relation needed to prove the numbers are in G.P.

Algebraic Equations

Algebraic equations are mathematical statements that use algebra to equate two expressions. They often include variables, constants, and arithmetic operations. Solving these equations provides specific values for the variables that make the equation true.

• Linear equations have variables with the power of one. They form straight-line graphs.
• Quadratic equations include a variable squared and form parabolic graphs.
• Systems of equations involve multiple equations solved together.

The task within the exercise was to manage algebraic equations involving harmonic means and solve them to establish a relationship. You rearranged and simplified them to show that certain terms form a G.P. Understanding and manipulating these equations are crucial in mathematical problem-solving.