Problem 95 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a, b, c\) are positive numbers in G.P. and log \(\left(\frac{5 c}{a}\right), \log \left(\frac{3 b}{5 c}\right)\) and \(\log \left(\frac{a}{3 b}\right)\) are in A.P. then \(a, b, c\) (A) form the sides of an equilateral triangle (B) form the sides of an isosceles triangle (C) form the sides of a right angled triangle (D) can not form the sides of a triangle

Step-by-Step Solution

Answer

(D) can not form the sides of a triangle.

Step 1: Conditions for Geometric Progression

Since \(a, b, c\) are in Geometric Progression, we know that \( \frac{b}{a} = \frac{c}{b} \). This implies that \( b^2 = ac \).

Step 2: Set up the Arithmetic Progression

Given that \( \log\left(\frac{5c}{a}\right), \log\left(\frac{3b}{5c}\right), \log\left(\frac{a}{3b}\right) \) are in Arithmetic Progression (A.P.), the difference between consecutive terms should be constant. Thus, \( \log\left(\frac{3b}{5c}\right) - \log\left(\frac{5c}{a}\right) = \log\left(\frac{a}{3b}\right) - \log\left(\frac{3b}{5c}\right) \).

Step 3: Simplify Logarithmic Expressions

This simplifies to: \[\log\left(\frac{3b}{5c}\right) = \frac{1}{2}\left(\log\left(\frac{5c}{a}\right) + \log\left(\frac{a}{3b}\right)\right)\]Combining logs we get: \[\log\left(\frac{3b}{5c}\right) = \frac{1}{2} \log\left(\frac{5c}{3b}\right)^{\frac{1}{2}} = 0\]Thus: \(\frac{3b}{5c} = 1\rightarrow 3b = 5c\)

Step 4: Solve the System of Equations

Using the results from Step 1 and Step 3, we solve the system of equations:1. \(b^2 = ac\)2. \(3b = 5c\)Substituting \(c = \frac{3}{5}b\) into equation 1 gives \(b^2 = a\left(\frac{3}{5}b\right) \rightarrow b = \frac{3a}{5}\).Thus, \(c = 3b/5 = 9a/25\).

Step 5: Check Triangular Inequalities

The sides \(a, b, c\) must satisfy the triangle inequalities:1. \(a + b > c\)2. \(a + c > b\)3. \(b + c > a\)Substituting \(b = \frac{3a}{5}, c = \frac{9a}{25}\), we check each inequality:1. \(a + \frac{3a}{5} > \frac{9a}{25}\) simplifies to \(\frac{8a}{5} > \frac{9a}{25}\), which holds true.2. \(a + \frac{9a}{25} > \frac{3a}{5}\) simplifies to \(\frac{34a}{25} > \frac{3a}{5}\), which is false.3. \(\frac{3a}{5} + \frac{9a}{25} > a\) simplifies to \(\frac{24a}{25} < a\), which holds true.Since one condition is false, they cannot form the sides of a triangle.

Key Concepts

Logarithmic Expressions in Arithmetic Progression

Logarithmic expressions are a way to represent numbers using powers or exponents. In this context, we are looking at three specific logarithmic expressions: \( \log\left(\frac{5c}{a}\right) \), \( \log\left(\frac{3b}{5c}\right) \), and \( \log\left(\frac{a}{3b}\right) \). These expressions are in arithmetic progression (A.P.), which means that the difference between the consecutive terms remains constant.

• A key property of logarithms used here is that \( \log(x/y) = \log(x) - \log(y) \).
• For the expressions to be in A.P., the difference \( \log\left(\frac{3b}{5c}\right) - \log\left(\frac{5c}{a}\right) \) must equal \( \log\left(\frac{a}{3b}\right) - \log\left(\frac{3b}{5c}\right) \).
• This simplifies by combining logs and using their properties to determine relationships between \( a, b, \) and \( c \).

Understanding how logarithmic differences relate in this context is crucial to solving the problem. This reveals hidden equations that lead to finding the specific proportional relationships of \( a, b, \) and \( c \).

Arithmetic Progression

Arithmetic progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is constant. This property becomes useful when evaluating sequences in various mathematical problems, including our exercise.

• In an A.P., if you know two terms, you can always find the third using the formula: \( x_{n+1} = x_n + d \), where \( d \) is the common difference.
• To verify if numbers form an A.P., calculate the difference between consecutive numbers. If these differences are the same, the numbers are in A.P.

Let's consider our problem. Using A.P. conditions on logs, once simplified, shows us relationships like \( 3b = 5c \), providing insights on the progression and connections among \( a, b, \) and \( c \). These relationships help determine not only the value structure but also potential characteristics when formed together, such as side lengths of geometric shapes.

Triangle Inequalities

Triangle inequalities state that for any triangle with sides \( a \), \( b \), and \( c \), certain conditions must be met. These conditions are essential in establishing whether a particular set of three numbers can be sides of a triangle.

• The inequality \( a + b > c \) ensures that two sides combined will always be greater than the third.
• Similarly, \( a + c > b \) and \( b + c > a \) confirm the need for the two other combinations of sides to add up to more than the third side.

When you substitute the values derived from the logarithmic analysis into these inequalities, some do not hold true, showing that \( a, b, \) and \( c \) cannot form a triangle. Specifically, \( a + c < b \) in our derived scenario, hence violating the triangle inequality condition, which is critical proof in solving these types of geometric questions.