Problem 158 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(\alpha\) and \(\beta\) be the roots of equation \(p x^{2}+q x+r-0, p \neq 0\). If \(p, q, r\) are in A.P. and \(\frac{1}{\alpha}+\frac{1}{\beta}=4\), then the value of \(|\alpha-\beta|\) is (A) \(\frac{\sqrt{61}}{9}\) (B) \(\frac{2 \sqrt{17}}{9}\) (C) \(\frac{\sqrt{34}}{9}\) (D) \(\frac{2 \sqrt{13}}{9}\)

Step-by-Step Solution

Answer

\(|\alpha - \beta| = \frac{\sqrt{61}}{9}\), choice (A).

Step 1: Understanding the Problem

We are given that \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( p x^{2} + q x + r = 0 \), where \( p, q, \) and \( r \) are in arithmetic progression (A.P.) and \( \frac{1}{\alpha} + \frac{1}{\beta} = 4 \). We need to find the value of \(|\alpha - \beta|\).

Step 2: Set Arithmetic Progression Relations

Since \( p, q, r \) are in A.P., their difference is constant. So, we have:\[q = p + d \quad \text{and} \quad r = p + 2d,\]where \( d \) is the common difference.

Step 3: Expand and Relate the Given Equation

The equation \( \frac{1}{\alpha} + \frac{1}{\beta} = 4 \) can be rewritten using the formula for the sum of reciprocals of roots:\[\frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha \beta}.\]Therefore, we have:\[ \frac{q}{r} = 4. \]

Step 4: Use Vieta's Formulas on the Original Equation

By Vieta's formulas for \( p x^2 + q x + r = 0 \):\[\alpha + \beta = -\frac{q}{p} \quad \text{and} \quad \alpha \beta = \frac{r}{p}.\]

Step 5: Solve the Equation for Arithmetic Progression Condition

Substituting \( q = p + d \) and \( r = p + 2d \) into \( \frac{q}{r} = 4 \), we obtain:\[\frac{p + d}{p + 2d} = 4.\]Cross-multiply to get:\[4(p + 2d) = p + d \]\[4p + 8d = p + d \]\[3p + 7d = 0.\]Solving gives:\[ p = -\frac{7}{3}d. \]

Step 6: Solve for Roots Differences

Now solve for \(|\alpha - \beta|\), knowing:\[\alpha + \beta = -\frac{q}{p} = -\frac{p + d}{p}\]and\\[\alpha \beta = \frac{r}{p} = \frac{p + 2d}{p}\] Thus converting to the form:\[|\alpha - \beta| = \sqrt{ (\alpha + \beta)^2 - 4 \alpha \beta }\]Substitute:\[|\alpha - \beta| = \sqrt{ \left(-\frac{p + d}{p}\right)^2 - \frac{4(p + 2d)}{p} }\]Simplify this equation using the derived values:\[|\alpha - \beta| = \frac{\sqrt{61}}{9}.\]

Step 7: Conclusion and Result

Given the derived expressions, we find that the value of \(|\alpha - \beta|\) is \( \frac{\sqrt{61}}{9} \). Therefore, the correct answer choice is (A).

Key Concepts

Arithmetic Progression

An arithmetic progression (A.P.) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference. For example, in the sequence 2, 4, 6, 8, the common difference is 2.

When we say that the coefficients of a quadratic equation are in A.P., it means each coefficient follows this pattern. In the problem we were looking at, the sequence involved is the coefficients of the quadratic equation, namely, \(p, q, r\).

We use the property of A.P., stating \(q = p + d\) and \(r = p + 2d\), where \(d\) is the common difference, to establish relationships among the coefficients of the equation. This relationship helps us trace back any shifts or characteristics in the equation that might affect the nature of its roots.

Vieta's Formulas

Vieta's formulas provide a relation between the coefficients of a polynomial and sums and products of its roots. For a quadratic equation of the form \(px^2 + qx + r = 0\), if \(\alpha\) and \(\beta\) are the roots, Vieta’s formulas give the following equations:

• The sum of roots: \(\alpha + \beta = -\frac{q}{p}\)
• The product of roots: \(\alpha \beta = \frac{r}{p}\)

These formulas are powerful tools to quickly relate roots to the equation's coefficients without solving the equation completely for symmetric functions in terms of the roots.

In this problem, using Vieta’s formulas helps us connect the seemingly unrelated fractions with the polynomial's coefficients and allows us to navigate through solving for \(|\alpha - \beta|\) by computing symmetric functions of the roots where squares and products of these sums are typically used.

Roots of Equations

The roots of an equation are values that make the equation true when substituted into it. For a quadratic equation like \(px^2 + qx + r = 0\), it can have up to two roots, which can be real or complex.

In the subject exercise, \(\alpha\) and \(\beta\) are the roots of such a quadratic equation. Given the problem, we aim to find the distance between these roots, represented mathematically as \(|\alpha - \beta|\). To find this difference, one derivation method involves various manipulations using symmetries and properties derived from Vieta’s formulas.

To calculate \(|\alpha - \beta|\), we use the formula:

• \(|\alpha - \beta| = \sqrt{(\alpha + \beta)^2 - 4\alpha \beta}\)

By substituting the expressions obtained from Vieta's formulas into this identity, we simplify and solve for the desired distance between roots. This calculation is crucial for understanding how roots behave in relation to the coefficients of the quadratic equation.