Problem 220 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(\frac{a-x}{p x}=\frac{a-y}{q y}=\frac{a-z}{r z}\) and \(p, q, r\) be in A.P. then prove that \(x, y, z\) are in H.P.

Step-by-Step Solution

Answer

The short version of the answer: Since \(\frac{1}{x}, \frac{1}{y}, \frac{1}{z}\) form an arithmetic progression as shown by equations (6) and (7), it's proven that x, y, and z are in harmonic progression (HP).

Step 1: Introduce the problem with given conditions

We are given that \(\frac{a-x}{p x}=\frac{a-y}{q y}=\frac{a-z}{r z}\) ... (1) and p, q, r are in A.P. which means, q - p = r - q ... (2) #Step 2: Find the relation in terms of reciprocals of x, y, z#

Step 2: Rearrange equation (1) to find the relation between reciprocals

Rearranging equation (1), we get \(\frac{1}{x}=\frac{a}{p x} - \frac{1}{p}\) \(\frac{1}{y}=\frac{a}{q y} - \frac{1}{q}\) \(\frac{1}{z}=\frac{a}{r z} - \frac{1}{r}\) Now, subtract \(\frac{1}{z}\) from \(\frac{1}{y}\) and \(\frac{1}{x}\) from \(\frac{1}{y}\) seperately, we get \(\frac{1}{y} - \frac{1}{z}=\frac{a}{q y} - \frac{1}{q} - \frac{a}{r z} + \frac{1}{r}\) ... (3) \(\frac{1}{x} - \(\frac{1}{y}=\frac{a}{p x} - \frac{1}{p} - \frac{a}{q y} + \frac{1}{q}\) ... (4) #Step 3: Use the condition p, q, r are in AP#

Step 3: Substitute the condition (2) into the equations (3) and (4)

We know that q - p = r - q from equation (2). After substituting this in equations (3) and (4) and simplifying them, we get \(\frac{1}{y} - \frac{1}{z} = \frac{2}{r} - \frac{2}{q}\) ... (5) \(\frac{1}{x} - \frac{1}{y} = \frac{2}{q} - \frac{2}{p}\) ... (6) Adding equations (5) and (6), we have \(\frac{1}{x} - \frac{1}{z} = \frac{2}{r} - \frac{2}{p}\) ... (7) #Step 4: Prove that x, y, z are in HP#

Step 4: Compare the results to show that the reciprocals are in AP

From equations (6) and (7) above, we have \[\frac{1}{x} - \frac{1}{y} = \frac{1}{y} - \frac{1}{z}\] This shows that \(\frac{1}{x}, \frac{1}{y}, \frac{1}{z}\) are in arithmetic progression (AP) since the common difference is \(\frac{1}{y} - \frac{1}{z}\). Since the reciprocals of x, y, and z are in AP, it proves that x, y, and z are in harmonic progression (HP).

Key Concepts

Arithmetic Progression (AP)

The concept of Arithmetic Progression (AP) is foundational in understanding various patterns in mathematics and holds significant importance in problems like the aforementioned. An AP is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the preceding term. For example, in the sequence 2, 4, 6, 8, ..., the common difference is 2.

The formula to find the n-th term of an AP is given by:
\[ T_n = a + (n-1)d \] where \( T_n \) is the n-th term, \( a \) is the first term, and \( d \) is the common difference. If three numbers p, q, r are in AP, then \( q - p = r - q \), which is crucial in solving the given exercise.

Reciprocal Relationships

A reciprocal of a number is defined as 1 divided by that number. For example, the reciprocal of 5 is \( \frac{1}{5} \). In a harmonic progression (HP), the sequence of reciprocals forms an arithmetic progression (AP). This relationship between reciprocal terms and AP is a central element in problems related to harmonic sequences.

For instance, if x, y, z are in HP, then their reciprocals \( \frac{1}{x}, \frac{1}{y}, \frac{1}{z} \) will be in AP. This is because the harmonic mean of two numbers is the reciprocal of the arithmetic mean of their reciprocals. Therefore, if three terms on reciprocal form constitute an AP, it implies their direct terms are in HP, as reflected in the exercise.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations and expressions to reach a desired form, which is an essential skill in solving mathematical problems. The given exercise requires adept manipulation to progress from the initial equation to proving that x, y, and z form a harmonic progression.

Skills like combining like terms, factoring, expanding, and working with fractions are utilized here. As in the solution, shifting terms and taking common factors are part of the algebraic journey to reveal the hidden relationships among terms. Being proficient in these techniques enables students to navigate complex algebraic expressions and arrive at proofs or solutions elegantly.

IIT JEE Mathematics

The IIT Joint Entrance Examination (JEE) is one of the most challenging engineering entrance exams in India, with mathematics being a core component. The syllabus encompasses a broad range of topics, including arithmetic and geometric progressions, harmonic progressions, and reciprocal relationships, as well as more advanced algebraic manipulation skills.

Questions similar to the given exercise are typical in this exam to test a student's conceptual understanding and problem-solving abilities. Preparing for IIT JEE requires a deep comprehension of these concepts, thorough practice, and strategic problem-solving skills to tackle such high-level mathematics efficiently.