Problem 136 - A Complete Resource Book in Mathematics for JEE Main

Question

In the following questions an Assertion (A) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason( \(\mathrm{R}\) ) is False (D) Assertion(A) is False, Reason(R) is True Assertion: If the roots of the equations \(x^{2}-b x+c=0\) and \(x^{2}-c x+b=0\) differ by the same quantity, then \(b+c\) is equal to \(-4\). Reason: If \(\alpha, \beta\) are the roots of the equation \(A x^{2}+\) \(B x+C=0\), then \(\alpha-\beta=\frac{\sqrt{B^{2}-4 A C}}{A}\)

Step-by-Step Solution

Answer

Both A and R are true; R is a correct explanation for A (Option A).

Step 1: Understanding the Problem

We need to evaluate the truth of the assertion and the reason given, and then determine if the reason correctly explains the assertion. The assertion states that if the roots of two quadratic equations differ by the same quantity, then \(b+c = -4\). The reason states a known formula for the difference of roots for a quadratic equation.

Step 2: Analyzing the Assertion

The first equation is \(x^2 - bx + c = 0\) with roots \(\alpha\) and \(\beta\). \(\)The second equation is \(x^2 - cx + b = 0\) with roots \(\gamma\) and \(\delta\). \(\)According to the assertion, \(\alpha - \beta = \gamma - \delta\). For roots \(\alpha, \beta\) we have the sum \(b\) and product \(c\). For roots \(\gamma, \delta\) we have sum \(c\) and product \(b\). Solve \(\alpha - \beta = \gamma - \delta\) using these relationships.

Step 3: Solving the Equations

Using the difference of roots formula, \(\alpha - \beta = \sqrt{b^2 - 4c}\) and \(\gamma - \delta = \sqrt{c^2 - 4b}\). \(\)Equating them, \(\sqrt{b^2 - 4c} = \sqrt{c^2 - 4b}\). \(\)Squaring both sides yields \(b^2 - 4c = c^2 - 4b\). Rearranging gives \(b^2 - c^2 = 4c - 4b\). \(\)Which can be rearranged as \((b - c)(b + c) = 4(c - b)\). \(\)Simplifying further: If \(b eq c\), \(b + c = -4\). Thus, the assertion is true.

Step 4: Verifying the Reason

The reason provides the formula \(\alpha - \beta = \frac{\sqrt{B^2 - 4AC}}{A}\) for general quadratic \(Ax^2 + Bx + C = 0\). This formula is indeed used to determine the difference of roots, confirming its correctness.

Step 5: Conclusion

Both the Assertion and the Reason are true. The formula given in the Reason is correctly used to explain why the difference of roots leads to \(b+c = -4\), since it directly helps derive the condition needed for the assertion. Thus, the reason is a correct explanation for the assertion.

Key Concepts

Roots of Quadratic Equations

Quadratic equations are polynomials of degree two, commonly expressed in the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are coefficients, where \( a eq 0 \). The roots of a quadratic equation are the values of \( x \) that satisfy the equation.
These roots can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]The symbol \( \pm \) indicates that there are generally two solutions or roots, often referred to as \(\alpha\) and \(\beta\). These roots may be real or complex, depending on the value of the discriminant \( b^2 - 4ac \). If the discriminant is positive, the roots are real and distinct; if it is zero, the roots are real and equal; and if it's negative, the roots are complex.

Understanding the nature and properties of these roots is crucial as they play a significant role in solving and analyzing quadratic equations. They not only tell us where the function intersects the x-axis if graphed, but also offer insights into the symmetry and vertex of the quadratic function.

Difference of Roots Formula

The difference of roots in a quadratic equation is an essential concept that helps in understanding the relationship between the coefficients and the roots. For any quadratic equation \( ax^2 + bx + c = 0 \), the difference between its roots \( \alpha \) and \( \beta \) can be calculated using the formula:\[ \alpha - \beta = \frac{\sqrt{b^2 - 4ac}}{a} \]This formula arises from the quadratic formula and helps in determining how far apart the roots are from each other, given the same set of coefficients.

In the exercise, two quadratic equations are involved, and the assertion revolves around having their roots differ by the same amount. By equating the differences from the two equations, we set a condition that results in a relation among the coefficients. For the particular case in this solution, it helps derive that \(b + c = -4\), highlighting how specifics of a quadratic equation can govern the constraints on its coefficients.

Quadratic Root Properties

The roots of a quadratic equation provide insight into several properties important for both theoretical and practical applications. Among these properties:

• The Sum of the Roots: For the quadratic equation \( ax^2 + bx + c = 0 \), the sum \( \alpha + \beta \) is given by \( -\frac{b}{a} \).
• The Product of the Roots: Similarly, the product \( \alpha \beta \) is \( \frac{c}{a} \).

These properties are derived from Vieta's formulas, which state relationships between the coefficients and roots without explicitly solving the quadratic equation.

In our specific exercise, the roots' properties play a central role in analyzing the assertion: the sum \( b \) and product \( c \) of the first equation, compared to the sum \( c \) and product \( b \) of the second. This consistent interchange helps derive conditions like \( b+c = -4 \). Understanding these properties allows us to navigate through more complex quadratic relationships and develop insights into the behavior and relationships between the roots and coefficients.