Problem 90 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a x^{2}+2 b x+c=0\) and \(a_{1} x^{2}+2 b_{1} x+c_{1}=0\) have a common root and \(\frac{a}{a_{1}}, \frac{b}{b_{1}}, \frac{c}{c_{1}}\) are in A.P., then \(a_{1}, b_{1}\), \(c_{1}\) are in (A) A.P. (B) G.P. (C) H.P. (D) None of these

Step-by-Step Solution

Answer

(A) A.P.

Step 1: Identify Relationship between Coefficients

We are given that \( \frac{a}{a_1}, \frac{b}{b_1}, \frac{c}{c_1} \) are in arithmetic progression (A.P.). This implies that the difference between consecutive terms is constant, i.e., \[ \frac{b}{b_1} - \frac{a}{a_1} = \frac{c}{c_1} - \frac{b}{b_1}. \]

Step 2: Express Relationship of the Common Root

Let \( \alpha \) be the common root of both quadratic equations. Then, we have from the given equations: \[ a \alpha^2 + 2b \alpha + c = 0 \] and \[ a_1 \alpha^2 + 2b_1 \alpha + c_1 = 0. \] This can also be written as \[ a_1 \alpha^2 + 2b_1 \alpha + c_1 = k(a \alpha^2 + 2b \alpha + c) = 0, \] which implies \( k = \frac{a_1}{a} = \frac{b_1}{b} = \frac{c_1}{c} \).

Step 3: Analyze the Relationship and Result

Since the terms \( \frac{a}{a_1}, \frac{b}{b_1}, \frac{c}{c_1} \) form an A.P., it implies that in reversing this expression to \( \frac{a_1}{a}, \frac{b_1}{b}, \frac{c_1}{c} \), these terms form a Harmonic Progression (H.P.). Thus, \( a_1, b_1, c_1 \) are in A.P.

Key Concepts

Common Root

A common root is a solution that satisfies two different equations simultaneously. Think of it as a common meeting point for two lines drawn on a graph. When dealing with quadratic equations, identifying a common root means finding a value of \( x \) that will make both equations true. For example, if \( \alpha \) is a common root of \( ax^2 + 2bx + c = 0 \) and \( a_1x^2 + 2b_1x + c_1 = 0 \), then substituting \( \alpha \) in place of \( x \) will simplify both equations to zero. This root connects the two quadratics, forming a link between their structures. Understanding this concept helps in solving algebraic expressions that hold unique connections.

Arithmetic Progression

Arithmetic Progression (A.P.) is a sequence of numbers in which the difference between consecutive terms is constant. For example, in the sequence 3, 5, 7, 9, each number increases by 2, making it an A.P. With the exercise in question, the coefficients \( \frac{a}{a_1}, \frac{b}{b_1}, \frac{c}{c_1} \) form an A.P. It implies that the ratio between successive coefficients grows or reduces by a common difference. This property is crucial in identifying specific relationships and patterns in sequences. Knowing about A.P. allows one to predict future terms, identify initial terms, and find missing elements in equations.

Harmonic Progression

Harmonic Progression (H.P.) is a bit different from arithmetic progression. A sequence is in H.P. if its reciprocals form an A.P. For instance, if the sequence 1, 1/2, 1/3 is considered, its reciprocal sequence 1, 2, 3 is an arithmetic progression, thus confirming it's an H.P. In our context, if \( \frac{a}{a_1}, \frac{b}{b_1}, \frac{c}{c_1} \) form an A.P., then \( \frac{a_1}{a}, \frac{b_1}{b}, \frac{c_1}{c} \) automatically form an H.P. Understanding this conversion between progressions is helpful in solving problems where transformation of values into more manageable forms is required. This concept aids in simplifying sequence evaluations in mathematical expressions.