Problem 105 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(b>a\), then the equation \((x-a)(x-b)-1=0\) has (A) both roots in \((-\infty, a)\) (B) one root in \((-\infty, a)\) and other in \((b, \infty)\) (C) both roots in \((b, \infty)\) (D) both roots in \([a, b]\)

Step-by-Step Solution

Answer

(C) both roots in \((b, \infty)\).

Step 1: Expand the Equation

First, expand the given equation \((x-a)(x-b)-1=0\). This results in \(x^2 - (a+b)x + ab - 1 = 0\).

Step 2: Analyze the Discriminant

For the quadratic equation \(ax^2 + bx + c = 0\), the discriminant is \(D = b^2 - 4ac\). Here, \(a=1\), \(b=-(a+b)\), and \(c=ab-1\). Calculate the discriminant: \[ D = (a+b)^2 - 4(ab-1) = a^2 + 2ab + b^2 - 4ab + 4 = a^2 - 2ab + b^2 + 4 = (a-b)^2 + 4 \] Since \(b > a\), \((a-b)^2\) is positive and \(D\) is always positive, indicating two distinct real roots.

Step 3: Identify Nature of Roots

Since the equation \((x-a)(x-b)-1=0\) has real and distinct roots, observe the symmetry of the quadratic equation roots around the midpoint \(\frac{a+b}{2}\). Given that \(f(x) ightarrow +\infty\) as \(x ightarrow \pm \infty\), and \(f(a) = -1 < 0\) and \(f(b) = -1 < 0\), this suggests a change in sign across the intervals \((-fty, a)\), \((a, b)\), and \((b, fty)\).

Step 4: Evaluate Intervals

The product of roots \(r1 \cdot r2 = ab - 1\) is less than \(a^2\) or \(b^2\) due to \(-1\) subtraction, thus both roots either fall into \([a, b]\) or are located in one end extending to infinity. By the previous function behavior and sign change analysis: - At \(x=a\) and \(x=b\), among all options, the only feasible interval without roots crossing midpoints defined by roots is \((b, fty)\). Thus, both roots are in interval \((b, \infty)\).

Key Concepts

Understanding the Discriminant

In quadratic equations, the discriminant is a crucial concept as it helps us determine the type of roots an equation will have. The discriminant is represented by the symbol \(D\), and it is calculated using the formula

• \(D = b^2 - 4ac\)

This formula is derived from the general form of a quadratic equation, \(ax^2 + bx + c = 0\). The value of the discriminant tells us several key characteristics about the roots of the equation:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root, also known as a repeated or double root.
• If \(D < 0\), the equation has no real roots and instead, the roots are complex or imaginary.

In the given exercise, we calculated the discriminant for the quadratic equation derived from \((x-a)(x-b)-1=0\). The result was \(D = (a-b)^2 + 4\). Since this is always positive, it indicates the presence of two distinct real roots because \((a-b)^2\) is positive when \(b > a\).
Thus, we confidently confirm that the equation has distinct real roots.

Identifying Real Roots

Real roots refer to the values of \(x\) that satisfy the quadratic equation \(ax^2 + bx + c = 0\) and are real numbers. These roots occur when the discriminant, \(D\), is either zero or positive. When a quadratic equation has two real roots, it means that the parabola defined by the equation intersects the x-axis at two points.
In this particular problem, we derived the equation from \((x-a)(x-b)-1=0\), expanding it to form \(x^2 - (a+b)x + (ab-1) = 0\). By analyzing the discriminant, we concluded it was positive, indicating two distinct real roots.
Understanding where these roots lie on the number line is crucial to solving the problem. By examining the behavior of the function at critical points and intervals, we determined that both real roots must reside in the interval \((b, \infty)\). This conclusion was drawn based on the behavior of the function and how it changes sign, affirming both roots are indeed real and within this specific range.

Exploring the Nature of Roots

The nature of roots of a quadratic equation refers to their characteristics based on the discriminant's value. It provides insight into whether the roots are real, distinct, or complex, and helps students anticipate how the graph of the quadratic will behave.
In the given exercise, we had to determine not only that the roots were real but also assess their positioning. The calculated positive discriminant indicates two distinct real roots, aligning with options like intervals they might reside in.

• The mid-point of roots symmetry at \(\frac{a+b}{2}\) suggests that the roots are spread symmetrically around this mid-point.
• The function's behavior, as it stretches toward \(+\infty\) or \(-\infty\), helps deduce intervals of negative or positive values.
• The important observations were values of \(f(x)\) approaching \(+\infty\) when \(x\) moves to either \(\pm \infty\) and being \(-1<0\) at points \(x=a\) and \(x=b\)

By examining function signs through the intervals, the consistent negative at critical points and becoming positive again in \((b, \infty)\), it was logical to conclude both roots are positioned beyond \(b\). This detailed understanding of root nature, along with discriminant analysis, supports those conclusions.