Problem 123 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the equation \(x^{2}+a^{2} x+b^{2}=0\) has two roots each of which exceeds a number \(c\), then (A) \(a^{4}>4 b^{2}\) (B) \(c^{2}+a^{2} c+b^{2}>0\) (C) \(-\frac{a^{2}}{2}>c\) (D) \(-\frac{a^{2}}{2}

Step-by-Step Solution

Answer

The correct answer is (C) \(-\frac{a^{2}}{2} > c\).

Step 1: Understand the Problem

We need to determine which condition ensures that the roots of the quadratic equation \(x^{2} + a^{2}x + b^{2} = 0\) are both greater than a given number \(c\). This involves understanding the relationships between the coefficients, the roots, and the threshold \(c\).

Step 2: Apply Vieta's Formulas

From Vieta's formulas, for a quadratic \(ax^2 + bx + c = 0\), the sum of the roots \(r_1 + r_2\) is equal to \(-\frac{b}{a}\) and the product \(r_1 \times r_2\) is \(\frac{c}{a}\). Here, \(-a^2\) is the sum of the roots and \(b^2\) is the product of the roots for our equation.

Step 3: Analyze Root Conditions

The roots exceed \(c\), meaning both roots \(r_1\) and \(r_2\) must satisfy \(r_1 > c\) and \(r_2 > c\). Therefore, the inequality for their sum \(r_1 + r_2 = -a^2\) compared to \(2c\) tells us that we need \(-a^2 > 2c\). This corresponds to \(c < -\frac{a^2}{2}\).

Step 4: Determine Correct Answer

From the analysis that both roots being greater than \(c\) implies \(-a^2 > 2c\), we match this condition to the given options. The corresponding option is \(\text{(C) } -\frac{a^{2}}{2} > c\).

Key Concepts

Vieta's Formulas

When working with quadratic equations, Vieta's formulas can be a real lifesaver. They help to easily find the sum and product of the roots without actually solving the equation. For any quadratic equation of the form \(ax^2 + bx + c = 0\), Vieta's formulas give us:

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

In the equation \(x^2 + a^2x + b^2 = 0\), we apply these formulas and find:

• Sum of the roots: \(r_1 + r_2 = -a^2\).
• Product of the roots: \(r_1 \times r_2 = b^2\).

These relationships are crucial because they allow us to directly connect the coefficients of the equation to the roots themselves. This is especially helpful when you are analyzing how the roots relate to any specific condition, such as exceeding a particular value \(c\).

Roots of the Equation

The roots of a quadratic equation are the solutions to the equation. These are the values of \(x\) where the equation equals zero. In this context, we have a quadratic equation \(x^2 + a^2x + b^2 = 0\). Solving for the roots gives us specific \(x\) values that satisfy the equation.
If you try to find these roots using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), you will get two distinct values: these are your \(r_1\) and \(r_2\). For our specific equation, these roots should be analyzed concerning an inequality.
Each root must be greater than a specific number, \(c\). This means that if both roots \(r_1\) and \(r_2\) are greater than \(c\), the condition can be mathematically represented through the sum and product of the roots, leading us to establish inequalities.

Inequalities in Quadratic Equations

In the context of finding roots and their properties, inequalities provide powerful insights. For this problem, the roots of the given quadratic equation \(x^2 + a^2x + b^2 = 0\) need to be greater than a certain number \(c\).
What does this inequality look like? Let's break it down:

• Both roots exceeding \(c\) tells us: \(r_1 > c\) and \(r_2 > c\).
• Add these two conditions: \(r_1 + r_2 > 2c\).
• According to Vieta's formulas, \(r_1 + r_2 = -a^2\), so \(-a^2 > 2c\).

This simple inequality \(-a^2 > 2c\) becomes pivotal because it translates the condition of roots being "greater than \(c\)" into a straightforward comparison involving the coefficients of the equation.
This leads to the conclusion \(c < -\frac{a^2}{2}\), which is a precise, mathematical representation of this relationship. This helps in deducing which condition among the given choices satisfies the problem's requirement.