Problem 25 - Algebra for JEE ADVANCED

Question

If \(a, b\), and \(c\) are in G.P. and \(x\) and \(y\), respectively, be arithmetic means between \(a, b\) and \(b, c\), then a. \(\frac{a}{x}+\frac{c}{y}=2\) b. \(\frac{a}{x}+\frac{c}{y}=\frac{c}{a}\) c. \(\frac{1}{x}+\frac{1}{y}=\frac{2}{b}\) d. \(\frac{1}{x}+\frac{1}{y}=\frac{2}{a c}\)

Step-by-Step Solution

Answer

The correct answer is c: \( \frac{1}{x} + \frac{1}{y} = \frac{2}{b} \).

Step 1: Understanding Geometric Progression

Given that \( a, b, c \) are in geometric progression, it means \( \frac{b}{a} = \frac{c}{b} \). This implies \( b^2 = ac \).

Step 2: Arithmetic Mean Definition

The arithmetic mean \( x \) between \( a \) and \( b \) is given by \( x = \frac{a + b}{2} \). Similarly, \( y \) is the arithmetic mean between \( b \) and \( c \), so \( y = \frac{b + c}{2} \).

Step 3: Expressing \( b \) in Terms of \( a \) and \( c \)

Since \( b^2 = ac \) from the G.P. property, it follows that \( b = \sqrt{ac} \).

Step 4: Calculate \( x \)

Substitute \( b = \sqrt{ac} \) into the equation for \( x \). Then \( x = \frac{a + \sqrt{ac}}{2} \).

Step 5: Calculate \( y \)

Similarly, substitute \( b = \sqrt{ac} \) into the equation for \( y \). Thus \( y = \frac{\sqrt{ac} + c}{2} \).

Step 6: Compute \( \frac{1}{x} + \frac{1}{y} \)

Substitute \( x \) and \( y \) into the fraction sum. Thus:\[ \frac{1}{x} + \frac{1}{y} = \frac{2}{a + \sqrt{ac}} + \frac{2}{\sqrt{ac} + c} \]

Step 7: Finding Common Denominator

The common denominator for \( \frac{1}{x} + \frac{1}{y} \) is \((a + \sqrt{ac})(\sqrt{ac} + c)\). Simplify using the identity \( (a+b)(a-b) = a^2-b^2 \):\[ = \frac{2(\sqrt{ac} + c) + 2(a + \sqrt{ac})}{a + c + 2\sqrt{ac}} = \frac{2a + 2c + 4\sqrt{ac}}{a + c + 2\sqrt{ac}} \]

Step 8: Simplifying the Expression

Simplifying the numerator and denominator, we get:\[ \frac{1}{x} + \frac{1}{y} = \frac{2(a+c+2\sqrt{ac})}{a+c+2\sqrt{ac}} \]The expression simplifies to \( \frac{2}{b} \) since \( b = \sqrt{ac} \).

Step 9: Conclusion

Thus, the correct option is c: \( \frac{1}{x} + \frac{1}{y} = \frac{2}{b} \).

Key Concepts

Arithmetic Mean

When we talk about the arithmetic mean, what we mean is that we're finding the average of two or more numbers. It's basically adding up the numbers and then dividing by the count of numbers. In the context of the problem, for the arithmetic mean between two numbers like \( a \) and \( b \), the formula becomes:

• For the arithmetic mean \( x \) between \( a \) and \( b \), \( x = \frac{a + b}{2} \).
• For the arithmetic mean \( y \) between \( b \) and \( c \), \( y = \frac{b + c}{2} \).

This concept is fundamental because it helps simplify and center values within a data set, making understanding relationships between numbers straightforward. In our problem, knowing the arithmetic means allows us to connect different values of the geometric progression elements, facilitating further algebraic manipulation.

Algebra

Algebra is the language of mathematics that lets us describe relationships and patterns using symbols and letters. In our exercise involving geometric progression, algebra allows us to express relationships concisely and solve for unknowns. Here's how algebra fits into our problem:

• Given that \( a, b, \) and \( c \) are in geometric progression, we start with the equation: \( \frac{b}{a} = \frac{c}{b} \). This relationship leads us to understand that \( b^2 = ac \), characterizing the G.P. setup.
• Throughout the problem, algebra allows us to derive expressions for \( b \) in terms of \( a \) and \( c \) such as \( b = \sqrt{ac} \).

Algebraic manipulation and simplification are key as we solve the equations and expressions in the steps, helping pinpoint relationships and ultimately deduce the solution for \( \frac{1}{x} + \frac{1}{y} = \frac{2}{b} \). This emphasizes that algebra is not just about numbers but is a toolkit for solving structured problems.

Problem Solving

Problem-solving in mathematics involves breaking down a complex question into manageable parts and relying on logical, step-by-step methods. In the given exercise, we've demonstrated problem-solving through structured steps:
First, we identified

• The essential characteristics of our sequence - that it's a geometric progression, ensuring the equation \( b^2 = ac \) holds.
• By calculating the arithmetic means \( x \) and \( y \), intermediary steps were outlined, simplifying the choice of variables using algebra and logic.

Then, solving equation by equation, we substituted values to simplify our expressions further.The approach to solve \( \frac{1}{x} + \frac{1}{y} = \frac{2}{b} \) primarily included identifying equivalent expressions and combining them intelligently to find the correct answer.
Moreover, being meticulous about finding the common denominator and performing simplifications demonstrated meticulous problem solving.Ultimately, the correct option "c" was found, emphasizing that each step logically flows from the last in the problem-solving path. Clear understanding and patience in execution define successful problem-solving in algebra.