Problem 97 - A Complete Resource Book in Mathematics for JEE Main 2017

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: Understand the Problem Statement

We need to find the nature of the sequence formed by \( \frac{a}{c}, \frac{b}{a}, \frac{c}{b} \) given the condition that the sum of the roots \( \alpha + \beta \) equals the sum of the squares of their reciprocals \( \frac{1}{\alpha^2} + \frac{1}{\beta^2} \). This is a property question related to sequences.

Step 2: Express the Given Condition

For a quadratic equation \( ax^2 + bx + c = 0 \), the roots are \( \alpha \) and \( \beta \). The sum of the roots \( \alpha + \beta = -\frac{b}{a} \). The sum of the squares of the reciprocals is \( \frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\alpha^2 + \beta^2}{\alpha^2 \beta^2} \). The condition given is \( \alpha + \beta = \frac{\alpha^2 + \beta^2}{\alpha^2 \beta^2} \).

Step 3: Use Algebraic Identities

Using the identity, \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \), substitute into the equation: \[-\frac{b}{a} = \frac{(\frac{b}{a})^2 - 2\frac{c}{a}}{\left(\frac{c}{a}\right)^2}\], \( (\alpha + \beta) \) can also be translated using Vieta's formulas.

Step 4: Solve the Expression

Substitute the expressions for \( \alpha + \beta \) and \( \alpha \beta \) from Vieta’s formulas, we rearrange the equation: \(-\frac{b}{a} = \frac{(\frac{b}{a})^2 - 2\left(\frac{c}{a}\right)}{\left(\frac{c}{a}\right)^2}\). Simplifying gives: \[ b^3 = 2abc \].

Step 5: Determine the Ratios

We now need to check if \( \frac{a}{c}, \frac{b}{a}, \frac{c}{b} \) form a specific sequence. Given \( b^3 = 2abc \), dividing the whole equation by \( abc \) gives \( \frac{b^3}{abc} = \frac{2abc}{abc} \) or \( \frac{b}{a} \cdot \frac{b}{c} = \frac{2a}{c} \). Rewrite: \( \frac{b^2}{ac} = 2 \), implies \( \frac{b}{a}, \frac{a}{c}, \frac{c}{b} \) are in harmonic progression.

Key Concepts

Roots of Quadratic Equation

Understanding the roots of a quadratic equation is fundamental in algebra. The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are coefficients with \(a eq 0\). The solutions for \(x\) are called the roots and can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] The equation has two roots, denoted as \(\alpha\) and \(\beta\). These roots can be real or complex numbers and represent the \(x\)-intercepts of the quadratic function on the Cartesian plane. Understanding the roots help in understanding the behavior of the quadratic equation, such as whether it touches or crosses the \(x\)-axis. These roots form the basis for many mathematical problems involving quadratic equations.

Vieta's Formulas

Vieta's formulas offer a shortcut to understanding the relationships between the coefficients of a polynomial and its roots. For a quadratic equation \(ax^2 + bx + c = 0\), the formulas are particularly useful:

• The sum of the roots \(\alpha + \beta\) is given by \(-\frac{b}{a}\). This shows how the coefficient \(b\) directly relates to the sum of the roots.
• The product of the roots \(\alpha \beta\) is \(\frac{c}{a}\). This showcases the role of the constant \(c\) in the product of the roots.

These relationships are derived directly from the equation, demonstrating the balance between the roots and the coefficients of the polynomial. Vieta's formulas can simplify many algebraic problems by bypassing the need to find roots explicitly, especially when only their sum or product is necessary.

Harmonic Progression

A harmonic progression (HP) involves terms that are reciprocals of an arithmetic progression. For example, if \(a, b, c\) form an arithmetic progression, then their reciprocals \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) form a harmonic progression. In our problem, determining that the sequence \(\frac{a}{c}, \frac{b}{a}, \frac{c}{b}\) is a harmonic progression involves understanding that these terms behave according to the rules of HP:

• The sequence satisfies the condition \(b^3 = 2abc\), which relates back to the harmonic mean.
• To verify HP, the condition \(2\left(\frac{b}{a}\right) = \left(\frac{a}{c}\right) + \left(\frac{c}{b}\right)\) must be satisfied.

This concept is another tool in identifying and categorizing mathematical sequences, expanding our ability to analyze complex algebraic patterns.