Problem 122 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a, b, c\) are positive rational numbers such that \(a>b>c\) and the quadratic equation \((a+b-2 c) x^{2}+\) \((b+c-2 a) x+(c+a-2 b)=0\) has a root in the interval \((-1,0,\), then (A) \(c+a<2 b\) (B) both roots of the given equation are rational (C) the equation \(a x^{2}+2 b x+c=0\) has both negative real roots (D) the equation \(c x^{2}+2 a x+b=0\) has both negative real roots

Step-by-Step Solution

Answer

(A) is the correct answer: \(c+a<2b\).

Step 1: Analyze the Quadratic Equation

Given the equation \((a+b-2c)x^2 + (b+c-2a)x + (c+a-2b) = 0\), we need to analyze the properties of this equation when one of its roots lies in the interval \((-1, 0)\). This implies that one root, say \(r\), satisfies \(-1 < r < 0\).

Step 2: Determine Constraints on Coefficients

By analyzing the quadratic equation given, the root range \(-1 < r < 0\) imposes certain constraints on the coefficients. It ensures that if \(r\) is a root, the constant term \((c+a-2b)\) must be positive because \(-r\) must cancel the effects of the other terms being negative in \(-1 < r < 0\). Thus \(c+a < 2b\).

Step 3: Evaluate Option (A)

The option \(c+a < 2b\) corresponds to our earlier determination that \((c+a-2b) > 0\). Thus, option (A) is a viable candidate based on the conditions set by the root.

Step 4: Evaluate Roots' Rationality for Options (B)

Since it is not explicitly given that the coefficients of the polynomial are rational numbers, we can't guarantee that the roots will be rational solely from the given conditions. So option (B) cannot be confirmed.

Step 5: Analyze Option (C) for Negative Roots

Consider the equation \(a x^2 + 2bx + c = 0\). The sum of the roots \(-\frac{2b}{a}\) and the product of the roots \(\frac{c}{a}\) should both be negative for both roots to be negative. Without more information, we cannot confirm these roots' negativity.

Step 6: Analyze Option (D) for Negative Roots

Consider the equation \(c x^2 + 2ax + b = 0\). Similarly, this option implies that \(-\frac{2a}{c}\) should be the negative sum of the roots and \(\frac{b}{c}\) the positive product. Verification without specific inequalities satisfying these conditions given is not possible solely from \(a > b > c\).

Step 7: Conclusion

Based on the analysis, option (A) \(c+a < 2b\) is the one consistent with the requirements posed by the constraint of having a root in \((-1, 0)\). This constraint ensures the positivity condition of the constant term in the equation.

Key Concepts

Root Properties

A quadratic equation is a polynomial equation of degree two, and its roots can tell us a lot about the equation's behavior. These roots, or solutions, are the values that make the equation equal to zero. For any quadratic equation in the form \(ax^2 + bx + c = 0\), the nature of its roots depends on the discriminant \(b^2 - 4ac\).

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If the discriminant is negative, the equation has two complex roots.

The interval constraint provided in the original exercise \((-1, 0)\) implies that the roots must not only be real (since they are within this real-number interval) but also help satisfy a range condition for the root being less than zero and greater than \(-1\). This particular set up pushes certain relationships of the coefficients so that their signs and magnitudes align with these restrictions, linking closely with specific inequality constraints.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient or fraction \(\frac{p}{q}\) of two integers, where \(p\) is an integer and \(q\) is a non-zero integer. In the context of quadratic equations, the coefficients need to be rational for us to potentially assert the rationality of the roots by examining factors like the discriminant.In the problem, we are told that \(a, b,\) and \(c\) are positive rational numbers. This means they can be rewritten as fractions with integer numerators and denominators. Rational roots theorem can be a handy tool here, suggesting that the potential rational roots of a polynomial are the factors of the constant term in relation to the leading coefficient. However, simply knowing the coefficients are rational does not guarantee that the roots will be rational unless additional specific conditions are satisfied.This challenge remains unresolved in the solution, as determining the roots' rationality solely based on the polynomial's rational coefficients when other conditions aren’t explicitly confirming can't be concretely achieved.

Inequality Constraints

Inequality constraints are crucial in mathematical problems to limit or define the possible solutions more precisely. In quadratic equations, they can define specific boundaries for the values of variables or parameters. In this exercise, an important inequality constraint that is defined is that one of the roots must lie in the interval \((-1, 0)\). This not only tells us about the nature of the root but also imposes constraints on the coefficients.When we say one root \(r\) must be between \(-1\) and \(0\), it means:

• The term \((c+a-2b)\) must be positive, ensuring the polynomial's behavior matches this root condition.
• Given the constraint of positive terms due to \(a, b, c\) being positive rational numbers, \(c + a < 2b\) thus becomes a binding condition deriving from the interval restraint on the roots.

Inequality manipulations and their fulfillment ensure the polynomial intersects the x-axis within the specified interval, dictating the values' necessary relationship to maintain this inequality in real scenarios.