Problem 96 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a, b, c\) are in G.P. and \(\log a-\log 2 b, \log 2 b-\log 3 c\) and \(\log 3 c-\log a\) are in A.P., then \(a, b, c\) are the sides of a triangle which is (A) right angled (B) acute angled (C) obtuse angled (D) None of these

Step-by-Step Solution

Answer

(C) obtuse angled

Step 1: Identify the conditions

We have two conditions: 1) The numbers \(a, b, c\) are in Geometric Progression (G.P.), and 2) The expressions \(\log a - \log 2b, \log 2b - \log 3c, \log 3c - \log a\) are in Arithmetic Progression (A.P.).

Step 2: Implication of G.P.

If \(a, b, c\) are in G.P., then there exists a common ratio \(r\) such that \(b = ar\) and \(c = ar^2\).

Step 3: Simplify the A.P. condition

For the sequence \(\log a - \log 2b, \log 2b - \log 3c, \log 3c - \log a\) to be in A.P., the differences between consecutive terms must be equal. Thus, we calculate: \((\log 2b - \log 3c) - (\log a - \log 2b) = (\log 3c - \log a) - (\log 2b - \log 3c)\).

Step 4: Solve the equations

Simplifying using the property \(\log m - \log n = \log\left(\frac{m}{n}\right)\), the first part of the equation is \(\log\left(\frac{2b}{3c}\right) - \log\left(\frac{a}{2b}\right) = \log\left(\frac{3c}{a}\right) - \log\left(\frac{2b}{3c}\right)\). This simplifies to: \[\log\left(\frac{4b^2}{3ac}\right) = \log\left(\frac{9c^2}{2ab}\right)\].

Step 5: Equate and solve for ratios

Since the above are equal, we have \(\frac{4b^2}{3ac} = \frac{9c^2}{2ab}\). Simplifying gives us \(8b^3 = 27c^2a\). Substituting \(b = ar\) and \(c = ar^2\) from the G.P. condition, hence \(8(ar)^3 = 27(ar^2)^2a\). Simplifying yields, \(8r^3 = 27r^4\), leading to \(8 = 27r\), or \(r = \frac{8}{27}\).

Step 6: Find expression for triangle sides

We have \(b = ar\) and \(c = ar^2\), substituting \(r = \frac{8}{27}\) results in \(b = a \times \frac{8}{27}\) and \(c = a \times \left(\frac{8}{27}\right)^2\). Evaluate: \(a^2 = b^2 + c^2 - 2bc \cos \theta\) leads to \((\frac{8}{27})^2 = 0.087791\).

Step 7: Nature of triangle

Given that \(b\) and \(c\) substantially less than \(a\), the property that \(a^2 \approx b^2 + c^2\) suggests this is an obtuse triangle, because the inequality \(a^2 > b^2 + c^2\) holds prominently.

Key Concepts

Arithmetic Progression

An Arithmetic Progression (A.P.) is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference. For example, in the sequence 3, 6, 9, 12, the common difference is 3. If the first term of an A.P. is denoted by \(a\) and the common difference by \(d\), then the \(n\)-th term can be expressed as \(a_n = a + (n-1)d\).

An interesting property of A.P. is that the average of any two terms in the sequence is equal to the term that is exactly midway between them. This property is pivotal in problems that require establishing that given expressions or terms form an arithmetic sequence.

• In the problem at hand, the differences \(\log a - \log 2b\), \(\log 2b - \log 3c\), and \(\log 3c - \log a\) are in A.P.
• This means \((\log 2b - \log 3c) - (\log a - \log 2b)\) should equal \((\log 3c - \log a) - (\log 2b - \log 3c)\) which helps us establish the necessary conditions.

Triangle properties

Triangles are fundamental shapes in geometry characterized by having three sides and three angles. Various properties allow us to classify triangles based on angles and side lengths. The main types of triangles by angle are right, acute, and obtuse:

• A **right triangle** has one angle equal to 90 degrees.
• An **acute triangle** has all three angles less than 90 degrees.
• An **obtuse triangle** has one angle greater than 90 degrees.

Knowledge of triangles is crucial in solving problems where you need to determine the nature of a triangle using its side lengths. Applying the Pythagorean theorem or its converse helps to identify the type of triangle:

• For a right triangle: \(a^2 = b^2 + c^2\).
• For an obtuse triangle: \(a^2 > b^2 + c^2\).
• For an acute triangle: \(a^2 < b^2 + c^2\).

In the problem you are working on, the goal is to determine if \(a, b, c\) form an obtuse triangle based on these principles. Using the given relationships and simplifications point towards \(a^2\) being greater than the sum of squares of \(b\) and \(c\), suggesting an obtuse triangle.

Logarithmic equations

Logarithmic equations involve logarithms of variables and require understanding the properties of logarithms for solving. The logarithm \( \log_b{x} \) represents the power to which a base \(b\) must be raised to yield \(x\).

For solving logarithmic equations, it's essential to keep the following properties in mind:

• The property \(\log_b{m} - \log_b{n} = \log_b{\left(\frac{m}{n}\right)}\) allows condensing the difference of two logs into a single expression.
• The change of base formula: \(\log_b{a} = \frac{\log_k{a}}{\log_k{b}}\) which simplifies logs with different bases.

In problems featuring both exponential and logarithmic terms, understanding the switch between exponential and logarithmic forms is crucial. The solution of the problem involves setting logarithmic expressions in arithmetic progression by using the property of logarithm differences to explore relationships among the variables.

In the given exercise's solution, the simplification and equating of logarithmic expressions such as \(\log\left(\frac{4b^2}{3ac}\right) = \log\left(\frac{9c^2}{2ab}\right)\) directly lead to solving for the unknowns applying properties of logarithms appropriately.