Problem 9 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(a, b, c\) be positive real numbers, such that \(b x^{2}+\) \(\left(\sqrt{(a+c)^{2}+4 b^{2}}\right) x+(a+c) \geq 0, \forall x \in R\), then \(a, b, c\) are in: (A) G.P. (B) A.P. (C) H.P. (D) None of these

Step-by-Step Solution

Answer

The numbers \(a, b, c\) are in Arithmetic Progression (A.P.).

Step 1: Analyze the given inequality

We are given the inequality \(b x^{2}+ \left(\sqrt{(a+c)^{2}+4 b^{2}}\right) x+(a+c) \geq 0\) for all real numbers \(x\). This is a quadratic equation in terms of \(x\) and for it to be non-negative for all \(x\), its discriminant must be zero or negative.

Step 2: Calculate the discriminant

The general form of a quadratic equation \(Ax^2 + Bx + C\) has a discriminant \(\Delta = B^2 - 4AC\). Plugging the coefficients into the formula, we have \( A = b, B = \sqrt{(a+c)^2 + 4b^2}, C = (a+c) \). Thus, the discriminant becomes \( \Delta = (\sqrt{(a+c)^2 + 4b^2})^2 - 4b(a+c) \).

Step 3: Simplify the discriminant expression

The expression \((\sqrt{(a+c)^2 + 4b^2})^2\) simplifies to \((a+c)^2 + 4b^2\). Therefore, the discriminant is \((a+c)^2 + 4b^2 - 4b(a+c)\). Simplifying further, we get \((a+c)^2 + 4b^2 - 4ab - 4bc\).

Step 4: Set the discriminant to zero

Since the quadratic must be non-negative for all \(x\), the discriminant should be zero: \((a+c)^2 + 4b^2 - 4ab - 4bc = 0\). Expanding, we have \(a^2 + 2ac + c^2 + 4b^2 - 4ab - 4bc = 0\).

Step 5: Rearrange terms and identify condition

Rearrange the expression: \(a^2 + c^2 + 2ac = 4ab + 4bc - 4b^2\). Notice that \(a, b, c\) are in Arithmetic Progression (A.P.) if \((a+c) = 2b\). Since we want the equation to hold as equality, substituting \(a+c = 2b\) satisfies it.

Key Concepts

Discriminant

In mathematics, the discriminant is a key concept when analyzing quadratic equations. Given the general quadratic form \(Ax^2 + Bx + C\), the discriminant is calculated using the formula \( \Delta = B^2 - 4AC \). The discriminant tells us about the nature of the roots the quadratic equation will have:

• If \(\Delta > 0\), the quadratic has two distinct real roots.
• If \(\Delta = 0\), there is exactly one real root, meaning the quadratic is a perfect square.
• If \(\Delta < 0\), the quadratic has no real roots, meaning its roots are complex numbers.

Understanding the discriminant is essential for solving problems involving quadratic inequalities and equations. In our original exercise, we needed the discriminant to be zero or negative to ensure the quadratic expression remains non-negative for all values of \(x\). This scenario is indicative of having either one real root or no real roots at all, thus guaranteeing the non-negativity of the quadratic expression.

Arithmetic Progression

An Arithmetic Progression (A.P.) is a sequence of numbers in which the difference between consecutive terms remains constant. This difference is known as the common difference. For example, in the sequence 3, 5, 7, 9, the common difference is 2. In the context of our problem, noticing the relationship between \(a, b,\) and \(c\) helps. For them to be in an arithmetic progression, they must satisfy the condition \(2b = a + c\). This implies:

• if \(a, b, c\) are successive terms, then \(b - a = c - b\)
• The middle term \(b\) is the average of \(a\) and \(c\)

Recognizing arithmetic progressions is vital in numerous mathematical contexts, as they often simplify the analysis and make relationships more evident, especially when dealing with quadratic inequalities and related conditions.

Positive Real Numbers

Positive real numbers are the set of real numbers that are greater than zero. They are fundamental in mathematics, serving numerous purposes across different areas such as analysis, geometry, and algebra. In inequalities, dealing with positive real numbers guarantees that any multiplicative inversion or logarithmic operation will yield defined results and remain meaningful. In our quadratic inequality exercise, specifying that \(a, b,\) and \(c\) are positive brings uniformity and helps regulate the expressions involved. This ensures that the solutions derived from setting up equations such as \(a + c = 2b\) make logical and mathematical sense. Working with positive real numbers gives clarity and precision, especially in forming and solving equations like quadratic forms, which depend on strict positivity to remain valid and interpretable.