Problem 100 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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 \(\quad\) [2003] (a) Arithmetic -Geometric Progression (b) Arithmetic Progression (c) Geometric Progression (d) Harmonic Progression.

Step-by-Step Solution

Answer

The terms \(\frac{a}{c}\), \(\frac{b}{a}\), and \(\frac{c}{b}\) form a geometric progression.

Step 1: Identify the relationships involving roots

For a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots is given by \(-\frac{b}{a}\) and the product of the roots is \(\frac{c}{a}\). We are given that the sum of the roots of the equation equals the sum of the squares of their reciprocals.

Step 2: Express reciprocals of roots

Let the roots be \(\alpha\) and \(\beta\). From Vieta's formulas, we know \(\alpha + \beta = -\frac{b}{a}\) and \(\alpha \beta = \frac{c}{a}\). The sums of squares of their reciprocals is \(\frac{1}{\alpha^2} + \frac{1}{\beta^2} = \frac{\alpha^2 + \beta^2}{(\alpha \beta)^2}\).

Step 3: Relate reciprocals to given condition

Given \(\alpha + \beta = \frac{1}{\alpha^2} + \frac{1}{\beta^2}\), we insert the known values for these quantities: \(-\frac{b}{a} = \frac{\alpha^2 + \beta^2}{\left(\frac{c}{a}\right)^2}\).

Step 4: Use identity for squares of roots

To find \(\alpha^2 + \beta^2\), use the identity \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2 \alpha \beta\). This becomes \(\left(-\frac{b}{a}\right)^2 - 2 \cdot \frac{c}{a} = \frac{b^2}{a^2} - \frac{2c}{a}\).

Step 5: Equate and solve for condition

With \(-\frac{b}{a} = \frac{\alpha^2 + \beta^2}{\left(\frac{c}{a}\right)^2}\), replace \(\alpha^2 + \beta^2\) with \(\frac{b^2 - 2ac}{a^2}\), giving us \(-\frac{b}{a} = \frac{b^2 - 2ac}{c^2/a^2}\), simplifying to \(-\frac{b}{a} = \frac{b^2 a - 2a^2 c}{c^2}\).

Step 6: Compare coefficients for progressions

Rewriting gives \(b c^2 = - a (b^2 - 2 a c )\) which leads us to relate coefficients \(\frac{a}{c}, \frac{b}{a}\), and \(\frac{c}{b}\). Substituting these into arithmetic series form, recognize these three expressions as satisfying geometric progression, where the cross multiplication of adjacent terms equal.

Key Concepts

Roots of Quadratic Equation

To understand quadratic equations, we first need to know what the 'roots' or 'solutions' of these equations are. A quadratic equation is of the form \( ax^2 + bx + c = 0 \). The values of \( x \) that satisfy this equation are called the roots. These roots are found using various methods such as factoring, using the quadratic formula, or completing the square.

For any quadratic equation, the sum of the roots \( \alpha + \beta \) is given by Vieta's formulas as \( -\frac{b}{a} \), and the product \( \alpha \beta \) is \( \frac{c}{a} \). These relationships are incredibly helpful for solving various problems, as they let us make connections between the coefficients of the equation and its roots without having to actually solve the equation every time.

Understanding these basics is crucial for exploring deeper concepts like Vieta's formulas and how they relate to mathematical progressions. By mastering the roots, you have the foundational tools needed to tackle more complex relationships in mathematics.

Progressions in Mathematics

Progressions are sequences of numbers that follow a specific pattern. There are different types of progressions, crucial among them being arithmetic, geometric, and harmonic.

• In an arithmetic progression (AP), each term after the first is obtained by adding a constant, known as the common difference \(d\), to the previous term. Example: 2, 5, 8, 11... is an AP where \(d = 3\).
• In a geometric progression (GP), each term after the first is obtained by multiplying the previous term by a constant, known as the common ratio \(r\). Example: 3, 6, 12, 24... is a GP where \(r = 2\).
• In a harmonic progression (HP), the reciprocals of the terms form an arithmetic progression. For example, 1, 1/2, 1/3, 1/4... (the reciprocals are 1, 2, 3, 4) forms an HP.

These progressions are fundamental in mathematics because they help us understand patterns and relationships within sets of numbers. Recognizing these sequences allows us to predict and calculate practically and precisely. In solving problems involving quadratic equations, identifying which type of progression certain coefficients form can reveal crucial insights about the solutions and their properties.

Vieta's Formulas

Named after the French mathematician François Viète, Vieta's formulas provide a remarkable way to link the coefficients of a polynomial to sums and products of its roots. For a quadratic equation \( ax^2 + bx + c = 0 \), Vieta's formulas tell us:

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

These relations are particularly powerful because they allow us to deduce significant information about the roots without solving the quadratic equation fully. For instance, knowing only the coefficients, we can predict whether the roots are real, imaginary, or rational.

In the exercise, understanding these formulas helped establish that \( \frac{a}{c}, \frac{b}{a}, \) and \( \frac{c}{b} \) form a geometric progression. This is because the cross multiplication characteristic of geometric progressions is satisfied, which originates from the product identities given by Vieta's relationships. By leveraging these formulas, we can uncover hidden symmetries and relationships within the equations, making them a potent tool in mathematics.