Problem 97 - A Complete Resource Book in Mathematics for JEE Main

Question

If the sum of the roots of the quadratic equation \(a x^{2}+\) \(b x+c=0\) is equal to the sum of the squares of their reciprocals, then \(\frac{a}{c}, \frac{b}{a}\) and \(\frac{c}{b}\) are in (A) arithmetic progression (B) geometric progression (C) harmonic progression (D) arithmetico-geometric progression

Step-by-Step Solution

Answer

(C) harmonic progression

Step 1: Understanding the Problem

We need to find a condition on the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \) such that when the sum of the roots is equal to the sum of the squares of their reciprocals, the ratios \( \frac{a}{c}, \frac{b}{a}, \frac{c}{b} \) form a particular progression.

Step 2: Express the Sum of the Roots

According to Vieta’s formulas, the sum of the roots \( \alpha \) and \( \beta \) of the equation \( ax^2 + bx + c = 0 \) is given by \( \alpha + \beta = -\frac{b}{a} \).

Step 3: Sum of the Squares of Reciprocals

The sum of the squares of the reciprocals of the roots is given by \( \left(\frac{1}{\alpha}\right)^2 + \left(\frac{1}{\beta}\right)^2 = \frac{\alpha^2 + \beta^2}{(\alpha \beta)^2} \).

Step 4: Relate the Two Given Conditions

We set the sum of the roots equal to the sum of the squares of their reciprocals: \( -\frac{b}{a} = \frac{\alpha^2 + \beta^2}{(\alpha\beta)^2} \).

Step 5: Express \(\alpha^2 + \beta^2\) in Terms of the Coefficients

Using the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \), we substitute to get: \( \frac{b^2}{a^2} - 2 \frac{c}{a} \).

Step 6: Set the Equations Equal

Equating these, we have \( -\frac{b}{a} = \frac{\frac{b^2 - 2ac}{a^2}}{\left(\frac{c}{a}\right)^2} \), which simplifies to \( b = 2c \).

Step 7: Determine the Type of Progression

With \( b = 2c \), consider the sequence \( \frac{a}{c}, \frac{b}{a}, \frac{c}{b} \). We have \( \frac{b}{a} = \frac{2c}{a}, \frac{c}{b} = \frac{c}{2c} = \frac{1}{2} \). The condition for a harmonic progression is satisfied as it relates to reciprocal adjustments.

Key Concepts

Sum of Roots

The sum of roots is a fundamental concept when dealing with quadratic equations. For a quadratic equation of the form \( ax^2 + bx + c = 0 \), the roots are the solutions to the equation. Vieta's formulas give us a direct relationship between the coefficients of the polynomial and its roots. In particular, the sum of the roots \( \alpha + \beta \) is given by the formula \( \alpha + \beta = -\frac{b}{a} \). This expression tells us that the sum of the roots depends on the ratio of the coefficient of the linear term \( b \) to the quadratic term \( a \). The negative sign indicates that if \( b \) is positive, the sum of the roots is negative, and vice versa. This relationship is crucial in simplifying complex expressions related to the roots.

Vieta’s Formulas

Vieta's formulas are a powerful tool in the field of polynomials, providing a bridge between the coefficients of a polynomial and its roots. For a quadratic equation \( ax^2 + bx + c = 0 \), Vieta's formulas state that:

• The sum of the roots \( \alpha + \beta = -\frac{b}{a} \) (as previously discussed).
• The product of the roots \( \alpha \beta = \frac{c}{a} \).

These formulas allow us to express complicated expressions involving the roots in simpler forms using the known coefficients. For instance, to find the sum of the squares of the roots, you'd first calculate \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \), and then substitute the values derived from Vieta’s formulas. This provides an easier way to understand and work with root-related equations without solving them directly.

Harmonic Progression

A harmonic progression is a sequence of numbers where the reciprocals are in arithmetic progression. If we have three terms \( a, b, c \) in harmonic progression, the reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) will form an arithmetic sequence. This property is sometimes used in problems involving ratios of polynomial coefficients.In the given exercise, it is concluded that the ratios \( \frac{a}{c}, \frac{b}{a}, \frac{c}{b} \) form a harmonic progression under the condition \( b = 2c \). This means the reciprocal of \( \frac{a}{c} \), \( \frac{b}{a} \), \( \frac{c}{b} \) when arranged accordingly, forms a straightforward arithmetic sequence. Identifying progressions like this helps in simplifying and solving polynomial-related problems, making the connections between algebra and arithmetic clearer.