Problem 146 - A Complete Resource Book in Mathematics for JEE Main

Question

The values of \(\alpha\) for which the sum of the squares of the roots of the equation \(x^{2}-(a-2) x-a-1=0\) assume the least value is (A) 1 (B) 0 (C) 3 (D) 2

Step-by-Step Solution

Answer

The values of \(\alpha\) for which the sum of the squares of the roots is minimal is \(\alpha = 1\), option (A).

Step 1: Identify the Roots

The given quadratic equation is \(x^2 - (a-2)x - (a+1) = 0\). Let the roots of the equation be \(p\) and \(q\). According to Vieta's formulas, \(p + q = a - 2\) and \(pq = -(a + 1)\).

Step 2: Express the Sum of Squares of the Roots

The sum of the squares of the roots \(p^2 + q^2\) can be expressed in terms of \(p + q\) and \(pq\) as follows: \(p^2 + q^2 = (p+q)^2 - 2pq\). Substitute the values from Vieta's formulas: \(p^2 + q^2 = (a-2)^2 - 2(-a-1)\).

Step 3: Simplify the Expression

Simplify the expression \((a-2)^2 + 2(a+1)\):\((a-2)^2 = a^2 - 4a + 4\).\(2(a+1) = 2a + 2\).So, \((a-2)^2 + 2(a+1) = a^2 - 4a + 4 + 2a + 2\).

Step 4: Combine and Simplify Further

Combine the terms:\(a^2 - 4a + 4 + 2a + 2 = a^2 - 2a + 6\). This expression \(a^2 - 2a + 6\) represents the sum of the squares of the roots \(p^2 + q^2\).

Step 5: Find the Value of \(\alpha\) for the Minimum

To find the \(\alpha\) that minimizes \(a^2 - 2a + 6\), complete the square or use the vertex form. The expression \(a^2 - 2a + 6\) can be rewritten as \((a - 1)^2 + 5\). The term \((a - 1)^2\) achieves its minimum value of 0 when \(a = 1\).

Step 6: Determine the Answer Option

Since the minimum value of \((a - 1)^2 + 5\) occurs at \(a = 1\), the correct answer is option (A), which corresponds to \(a = 1\).

Key Concepts

Vieta's Formulas

Vieta's formulas are a powerful tool in the realm of quadratic equations. These formulas provide a link between the coefficients of a polynomial and sums and products of its roots. They are derived from the fundamental properties of algebra.

For any quadratic equation of the form \(x^2 + bx + c = 0\), Vieta's formulas state:

• The sum of the roots \(p + q = -b\).
• The product of the roots \(pq = c\).

In the given exercise, our quadratic equation is \(x^2 - (a-2)x - (a+1) = 0\). Applying Vieta's formulas, we find:

• Sum of the roots \(p + q = a - 2\).
• Product of the roots \(pq = -(a+1)\).

These relationships allow us to transform and manipulate roots into various expressions, such as the sum of squares, crucial for solving the problem discussed.

Sum of Squares of Roots

The sum of squares of the roots of a quadratic equation is often required to understand the nature of the roots. Given roots \(p\) and \(q\) of a quadratic equation, you might need \(p^2 + q^2\). Luckily, this can be expressed in terms of \(p + q\) and \(pq\) using the expansion:

\[ p^2 + q^2 = (p + q)^2 - 2pq \]This formula simplifies the calculation by utilizing Vieta’s outcomes. For our equation, substituting the known formulas, we have:

• \(p^2 + q^2 = (a-2)^2 - 2(-a-1)\).
• Which becomes: \((a-2)^2 + 2(a+1)\).

Simplifying further, we see:

• \((a-2)^2 = a^2 - 4a + 4\)
• \(2(a+1) = 2a + 2\)

Thus:

• \(p^2 + q^2 = a^2 - 2a + 6\).

Being able to express the sum of squares this way is essential to solving many algebraic problems efficiently.

Minimizing Quadratic Expression

To understand the behavior of quadratic expressions, such as minimizing them, it's important to recognize the form of the function. In this case, we want to minimize the expression \(a^2 - 2a + 6\).

The strategy to simplify and minimize such expressions usually involves completing the square, which groups terms into a perfect square plus a constant. This method reveals the vertex of the parabola described by the quadratic expression, indicating the minimum or maximum point. Let's see how it works for our expression:

• Start with \(a^2 - 2a + 6\).
• Rearrange it as \((a - 1)^2 + 5\).

This transformation shows the expression has its minimum when \((a - 1)^2\) equals zero, because any square is non-negative. Thus, the whole expression achieves its minimum value when \(a = 1\), making the minimal value \(5\).

Understanding these practices allows you to effectively approach optimization problems in quadratic expressions, crucial for algebra and calculus developments.