Problem 38 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(a, b, c\) be real numbers, \(a \neq 0 .\) If \(\alpha\) is a root of \(a^{2} x^{2}+b x+c=0, \beta\) is a root of \(a^{2} x^{2}-b x-c=0\) and \(0<\alpha<\beta\), then the equation \(a^{2} x^{2}+2 b x+2 c=0\) has a root \(\gamma\) that always satisfies (A) \(\gamma=\frac{\alpha+\beta}{2}\) (B) \(\gamma=\alpha+\frac{\beta}{2}\) (C) \(\gamma=\alpha\) (D) \(\alpha<\gamma<\beta\)

Step-by-Step Solution

Answer

The root \(\gamma\) satisfies \(\alpha < \gamma < \beta\) (D).

Step 1: Identify the given quadratic equations

We are given two quadratic equations: \(a^2x^2 + bx + c = 0\) with root \(\alpha\), and \(a^2x^2 - bx - c = 0\) with root \(\beta\). We need to find the relationship between these roots.

Step 2: Use the relationship between coefficients and roots

For the first equation, by Vieta's formulas, we know: \(\alpha_1 + \alpha_2 = -\frac{b}{a^2}\) and \(\alpha_1 \alpha_2 = \frac{c}{a^2}\). Similarly, for the second equation, \(\beta_1 + \beta_2 = \frac{b}{a^2}\) and \(\beta_1 \beta_2 = \frac{-c}{a^2}\). Since we are given \(0 < \alpha < \beta\), the equations suggest some peculiarities about the nature of the roots.

Step 3: Investigate the new equation

The new equation \(a^2x^2 + 2bx + 2c = 0\) transforms the previous systems. Using Vieta's formulas again, we find the sum of roots \( -\frac{2b}{a^2}\) and the product of roots \(\frac{2c}{a^2}\).

Step 4: Determine the root behavior of the new equation

We want to look at how the roots \(\gamma\) fit into the arrangement. Since \(0 < \alpha < \beta\), and considering the shift in the coefficients, this introduces a central root \(\gamma\) that is likely positioned between \(\alpha\) and \(\beta\). This suggests that \(\gamma\) should fall into a position such that \(\alpha < \gamma < \beta\), similar to an average.

Key Concepts

Vieta's formulas

Vieta's formulas are a powerful tool for understanding the relationships between the roots and coefficients of a polynomial equation.
These formulas make it easy to find the sum and product of the roots without solving the equation completely.
For a quadratic equation, specifically in the form of \(ax^2 + bx + c = 0 \), Vieta's formulas help us by providing these succinct relationships between the coefficients and the roots:

• The sum of the roots: \( r_1 + r_2 = -\frac{b}{a} \)
• The product of the roots: \( r_1 \cdot r_2 = \frac{c}{a} \)

These relationships result from comparing the coefficients of the polynomial to an expanded product of binomials formed by the roots.
In our context, Vieta's formulas assist in discerning the characteristics of the roots \(\alpha\) and \(\beta\) of our initial quadratic equations.
They reveal how these roots relate directly to the coefficients, aiding in further exploration of their properties.

Roots of Quadratic Equations

The roots or solutions of a quadratic equation like \(ax^2 + bx + c = 0\) are values of \(x\) which make the equation true.
To find these roots, you typically use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula helps in understanding how changes in coefficients \(a\), \(b\), and \(c\) influence the solutions.
In simpler terms:

• "\(-b\)" acts as a shift, moving the roots along the number line.
• "\(b^2 - 4ac\)" or the discriminant, determines the nature of roots (real or imaginary).
• "4a" beneath the square root affects the breadth around the average, affecting spread.

In the given exercise, \(\alpha\) and \(\beta\) arise from similar quadratic identities, yet with opposite signs for \(b\) and \(c\).
They explore how such adjustments manifest between real roots, aligning excellently with the introduction of an intermediate root \(\gamma\) within a new equation.

Relationship Between Roots and Coefficients

Understanding the relationship between roots and coefficients allows us to predict and explore how roots will behave under different algebraic manipulations.
With a quadratic, the nature and position of the roots are deeply connected to its coefficients.
For example:

• Root positions are critically linked to the sign and value of \(b\).