Problem 100 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(a\) be a fixed real number such that \(\frac{a-x}{p x}=\frac{a-y}{q y}=\frac{a-z}{r z}\)If \(p, q, \mathrm{r}\) are in A.P. then \(x, y, z\) are in (A) A.P. (B) G.P. (C) H. P (D) None of these

Step-by-Step Solution

Answer

x, y, z are in H.P.

Step 1: Understand the Equation Condition

We are given that \( \frac{a-x}{p x} = \frac{a-y}{q y} = \frac{a-z}{r z} = k \), where \( k \) is a constant. This implies three equations based on equality to \( k \).- \( a - x = k p x \)- \( a - y = k q y \)- \( a - z = k r z \)

Step 2: Express x, y, and z in terms of p, q, r

Using the equation expressions, re-arrange to find the variables explicitly:- For \( x \): \( a = x(1+k p) \) gives \( x = \frac{a}{1+k p} \)- For \( y \): \( a = y(1+k q) \) gives \( y = \frac{a}{1+k q} \)- For \( z \): \( a = z(1+k r) \) gives \( z = \frac{a}{1+k r} \)

Step 3: Relate p, q, r as an Arithmetic Progression

Since \( p, q, r \) are in Arithmetic Progression, we know:\( 2q = p + r \).This relationship will help us explore the pattern of \( x, y, z \).

Step 4: Determine if x, y, z are in A.P., G.P., or H.P.

To find the pattern for \( x, y, z \), let's convert them into a common form:Given \( x = \frac{a}{1+k p}, y = \frac{a}{1+k q}, z = \frac{a}{1+k r} \), observe:\( x, y, z \) represent the reciprocals of the expressions on the denominator with common numerators. If \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) are in Arithmetic Progression, then \( x, y, z \) are in Harmonic Progression.

Key Concepts

Arithmetic Progression

Arithmetic Progression, often abbreviated as A.P., is a sequence of numbers in which the difference between any two consecutive terms is always the same. This difference is known as the common difference.

• For example, in the sequence 2, 4, 6, 8, ..., the common difference is 2.
• An arithmetic sequence can be represented as: \[ a, a+d, a+2d, a+3d, \ldots \], where \( a \) is the first term and \( d \) is the common difference.
• Understanding A.P. is essential when solving problems that involve patterns or sequences where addition or subtraction occurs regularly.

These principles simplify the process of determining whether a sequence of numbers is in Arithmetic Progression by checking the consistent difference between each pair of terms.

Equations

Equations are mathematical statements that assert the equality of two expressions. They play a fundamental role in conveying relationships between numbers and variables.

• An equation will have an equal sign (\(=\)) which means both sides of the equation are balanced.
• For the equation \( ax + b = c \), solving for \(x\) involves isolating the variable by applying inverse operations.
• Equations can be simple linear forms or more complex involving quadratic or polynomial expressions.

In mathematics, particularly when dealing with progressions or sequences, equations help translate the relationships between numbers and unknown values into solvable problems.

Reciprocals

Reciprocals are the inverse of a number with respect to multiplication, meaning that when a number is multiplied by its reciprocal, the result is 1. Knowing about reciprocals is essential for understanding harmonic progressions and certain algebraic manipulations.

• The reciprocal of a number \( x \) is \( \frac{1}{x} \).
• For example, the reciprocal of 3 is \( \frac{1}{3} \), and the reciprocal of \( \frac{1}{2} \) is 2.
• Reciprocals are particularly important in solving equations where division by a variable or constant is involved.

Understanding how reciprocals function helps in solving equations for Harmonic Progressions, where the reciprocal nature of terms is a key characteristic of the sequence.

Real Numbers

Real numbers comprise a vast set of numbers encompassing all the numbers that can be found on the number line. This set includes both rational and irrational numbers, essentially every number you can think of.

• Real numbers include integers, fractions, and decimal numbers with infinite non-repeating sequences.
• Examples include 5, -3, 0, \( \frac{1}{2} \), and \( \sqrt{2} \).
• They are crucial in mathematics because they allow us to perform operations and calculations with practically any hyper-imaginable number.

In progressing problems, such as this one involving Harmonic Progression, recognizing what constitutes a real number allows for a smooth approach to number operations in the given context.