Problem 83 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a, b, c\) are in G.P., then the equations \(a x^{2}+2 b x+c=\) 0 and \(d x^{2}+2 e x+f=0\) have a common root if \(\frac{d}{a}, \frac{e}{b}\), \(\frac{f}{c}\) are in (A) H.P. (B) G.P. (C) A.P. (D) None of these

Step-by-Step Solution

Answer

The ratios form a G.P., so the answer is (B) G.P.

Step 1: Understanding Geometric Progression (G.P.)

In a geometric progression (G.P.), the ratio of successive terms is constant. If \(a, b, c\) are in G.P., then \( \frac{b}{a} = \frac{c}{b} = r \), where \(r\) is the common ratio. Hence, \(b = ar\) and \(c = ar^2\).

Step 2: Express the Common Root Condition

If two quadratic equations have a common root, then the ratios of their corresponding coefficients satisfy specific conditions depending on the sequences. These conditions often lead to some common mathematical series when verified.

Step 3: Formulating the Problem

Given the equations: \( ax^2 + 2bx + c = 0 \) and \( dx^2 + 2ex + f = 0 \), and knowing \(a, b, c\) are in G.P., our goal is to check how the ratios \(\frac{d}{a}, \frac{e}{b}, \frac{f}{c}\) form a series.

Step 4: Analyzing Ratios Corresponding to G.P.

Using the fact that \(a, b, c\) are in G.P., we check if \( \frac{d}{a}, \frac{e}{b}, \frac{f}{c} \) follow a similar mathematical series as G.P. Suppose \(\frac{d}{a}, \frac{e}{b}, \frac{f}{c} \) are also in G.P., this implies \( \frac{e}{b} \div \frac{d}{a} = \frac{f}{c} \div \frac{e}{b} \).

Step 5: Identifying the Correct Series

Since, given the condition \( \frac{d}{a}, \frac{e}{b}, \frac{f}{c} \) are such that they manifest similar properties to G.P. when compared alongside known mathematical series, they establish a G.P. as a similar pattern follows between successive divisions.

Step 6: Conclusion

Therefore, the given ratios \(\frac{d}{a}, \frac{e}{b}, \frac{f}{c}\) are in Geometric Progression (G.P.), as they maintain a constant ratio between successive terms.

Key Concepts

Geometric Progression (G.P.)

In Algebra, a Geometric Progression (G.P.) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, if we have three terms in a G.P. such as \(a, b, c\), the relationship can be expressed as follows:

• \( \frac{b}{a} = \frac{c}{b} = r \)

This means that if you take the second term \(b\) and divide it by the first term \(a\), you should get the same result as when you divide the third term \(c\) by the second term \(b\). This result is the common ratio \(r\).
For the given terms \(a, b, c\) we can write:

• \(b = ar\)
• \(c = ar^2\)

Thus, understanding G.P. helps identify how terms are related to each other when expressed in this sequential manner.

Quadratic Equations

Quadratic equations are an essential concept in Algebra, characterized by the term \( x^2 \) as the highest power of the variable \(x\). A standard form of a quadratic equation is \( ax^2 + bx + c = 0 \), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The solutions to the quadratic equation are given by the formula:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \(b^2 - 4ac\) is known as the discriminant which determines the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If it is negative, the roots are complex and occur as conjugates.

Quadratic equations often interact with sequences like G.P. by establishing relationships between their coefficients and special series conditions.

Common Root Conditions

The condition of common roots in the context of quadratic equations refers to the situation where two equations have at least one solution in common. If we consider two quadratic equations:

• \(ax^2 + 2bx + c = 0\)
• \(dx^2 + 2ex + f = 0\)

For these equations to have a common root, the coefficients must satisfy specific criteria. Particularly, the ratios of their coefficients \( \left(\frac{d}{a}, \frac{e}{b}, \frac{f}{c}\right) \) can sometimes form notable series.
In the mentioned problem, it's discovered that if these ratios form a Geometric Progression, it implies that the two quadratic equations can have a common root under given conditions. This relates back to how sequences like G.P. influence the structure and solution of quadratic equations. The analysis of such conditions helps simplify the understanding and solving of problems involving multiple quadratic equations.