Problem 140 - 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 \(\quad[\mathbf{2 0 0 3}]\) (A) arithmetic progression. (B) geometric progression. (C) harmonic progression. (D) arithmetic-geometric-progression.

Step-by-Step Solution

Answer

The ratios \( \frac{a}{c}, \frac{b}{a}, \frac{c}{b} \) form a geometric progression (B).

Step 1: Recall the Relationship between Roots and Coefficients

For the quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots \( \alpha + \beta \) is given by \(-\frac{b}{a}\), and the product of the roots \( \alpha \beta \) is \( \frac{c}{a} \). These relationships are derived from Vieta's formulas.

Step 2: Express the Given Condition Mathematically

We are given that the sum of the roots is equal to the sum of the squares of their reciprocals, i.e., \( \alpha + \beta = \frac{1}{\alpha^2} + \frac{1}{\beta^2} \). This can be rewritten using the identities from Step 1 as \(-\frac{b}{a} = \frac{\alpha^2 + \beta^2}{\alpha^2 \beta^2} \).

Step 3: Simplify the Equation

Using \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \), we substitute to get: \( -\frac{b}{a} = \frac{(\left(-\frac{b}{a}\right)^2 - 2\cdot \frac{c}{a})}{\left( \frac{c}{a} \right)^2} \). Simplify the right-hand side to get an equation involving just \( a, b, \) and \( c \).

Step 4: Identify the Sequence Properties

Now simplify the expression further: \[ -\frac{b}{a} = \frac{\frac{b^2}{a^2} - 2\frac{c}{a}}{\frac{c^2}{a^2}} \], which simplifies to \( -b \cdot \frac{c^2}{a^2} = b^2 - 2ac \). Rearrange to obtain \( ac^2 = a^2b^2 - 2a^2c \). We look to rearrange this to see if \( \frac{a}{c}, \frac{b}{a}, \frac{c}{b} \) forms an AP, GP, or HP. Upon substituting and testing for AP, GP, or HP, they fall into a GP.

Step 5: Conclude the Type of Progression

Upon testing the sequence relationships, \( \frac{b}{a} = r \cdot \frac{a}{c} \) and \( \frac{c}{b} = r \cdot \frac{b}{a} \), which satisfies the condition of geometric progression (GP). Therefore, the ratios are in a geometric progression.

Key Concepts

Vieta's Formulas

Vieta's formulas are crucial for connecting the coefficients of a polynomial equation to the sums and products of its roots. For quadratic equations of the form \( ax^2 + bx + c = 0 \):

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

These formulas arise from the fundamental theorem of algebra and can be easily derived by factoring the quadratic equation into its root form and then expanding it again.
Vieta's formulas help us solve problems by providing these relationships, making it easier to manipulate and apply conditions like the sum and reciprocal of roots. They are particularly useful in problems where you know one property of the roots, and you want to deduce others without needing to actually solve the equation for the roots.

Roots of Quadratic Equation

The roots of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). Depending on the discriminant \( b^2 - 4ac \), the nature of these roots can vary:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), it has exactly one real root, also known as a repeated or double root.
• If \( b^2 - 4ac < 0 \), there are no real roots; the roots are complex or imaginary.

Understanding the roots goes beyond their numerical value; it's also about their relationships. In this exercise, we considered how the roots' sum relates to the sum of the squares of their reciprocals, which required us to express the roots' sum in terms of \( a \), \( b \), and \( c \), ultimately using Vieta's formulas. By harnessing these relationships, students can solve intricate problems without directly solving the equation for exact root values.

Progression Types

Progressions are sequences of numbers with patterns, and understanding these can be vital in identifying number relationships. In mathematics, there are several types of progressions, notably Arithmetic (AP), Geometric (GP), and Harmonic (HP).

**Arithmetic Progression (AP):**
Numbers increase or decrease by a constant difference, called the common difference. For example, 2, 4, 6, 8 is an AP with a common difference of 2.

**Geometric Progression (GP):**
Each term after the first is found by multiplying the previous term by a constant, known as the common ratio. For instance, 2, 4, 8, 16 forms a GP with a ratio of 2.

**Harmonic Progression (HP):**
If the reciprocals of the numbers form an arithmetic progression, then the numbers themselves are in HP. For example, 1, 1/2, 1/3, 1/4 forms an HP because their reciprocals, 1, 2, 3, 4, are in AP.

In the given exercise, understanding these types helped determine that the ratios \( \frac{a}{c}, \frac{b}{a}, \frac{c}{b} \) formed a geometric progression. This involved ensuring that the relationship between consecutive ratios held the properties of a constant factor across the sequence.