Problem 210 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(\alpha, \beta\) be real numbers and \(z\) a complex number. If \(z^{2}+\alpha z+\beta=0\) has two distinct roots on the line \(\operatorname{Re}(z)=1\), then it is necessary that (A) \(\beta \in(-1,0)\) (B) \(|\beta|=1\) (C) \(\beta \in(1, \infty)\) (D) \(\beta \in(0,1)\)

Step-by-Step Solution

Answer

(C) \(\beta \in(1, \infty)\)

Step 1: Understand the Roots on Line

Given that the roots of the equation \(z^2 + \alpha z + \beta = 0\) lie on the line \(\operatorname{Re}(z) = 1\), it means that the real part of both roots is 1. Let’s express the roots in terms of complex numbers: \(z_1 = 1 + yi\) and \(z_2 = 1 + xi\), where \(x\) and \(y\) are real numbers.

Step 2: Use Vieta’s Formulas

By Vieta's formulas, the sum of the roots \((z_1 + z_2)\) is equal to \(-\alpha\). Replacing the roots, we get \(z_1 + z_2 = (1 + yi) + (1 + xi) = 2 + (x + y)i\). Thus, \(-\alpha = 2 + (x + y)i\). Since \(\alpha\) is real, \(x + y = 0\). So \(y = -x\).

Step 3: Determine the Product Equation

By Vieta's formulas, the product of the roots \(z_1z_2 \) equals \(\beta\). Substitute \(z_1 = 1 + yi\) and \(z_2 = 1 - yi\) into the product: \((1 + yi)(1 - yi) = 1 - (yi)^2 = 1 + y^2\). Therefore, \(\beta = 1 + y^2\).

Step 4: Analyze the Distinct Roots Condition

The condition that \(z_1\) and \(z_2\) are distinct implies \(y eq 0\). When \(y eq 0\), \(y^2 > 0\), which means \(1 + y^2 > 1\). Hence, \(\beta > 1\).

Step 5: Choose the Correct Answer

From the analysis, it is clear that \(\beta = 1 + y^2\) where \(y^2 > 0\) because the roots are distinct. Therefore, \(\beta\) must be greater than 1, which leads to the conclusion \(\beta \in(1, \infty)\). The correct answer is (C).

Key Concepts

Vieta's Formulas

Vieta’s formulas are a powerful tool in algebra that relate the coefficients of a polynomial to sums and products of its roots. For a quadratic equation like \( ax^2 + bx + c = 0 \), Vieta's formulas establish that:\

• The sum of the roots \( \alpha + \beta \) is equal to \( -\frac{b}{a} \).
• The product of the roots \( \alpha \cdot \beta \) is equal to \( \frac{c}{a} \).

In the exercise, the equation \( z^2 + \alpha z + \beta = 0 \) is considered, which is a standard form of a quadratic equation with \( a = 1 \). Therefore, Vieta's formulas simplify to:\

• \( \alpha = -(z_1 + z_2) \)
• \( \beta = z_1 \cdot z_2 \)

These relationships help determine the roots' properties without actually solving for the roots directly, making them particularly useful when dealing with complex roots.

Quadratic Equations

Quadratic equations are algebraic expressions of the form \( ax^2 + bx + c = 0 \). These equations are central in algebra because they represent problems involving parabolic relationships. A quadratic equation typically has two solutions or roots, which can be real or complex.

Roots of a quadratic equation are found using the quadratic formula:\\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

• When \( b^2 - 4ac > 0 \), there are two distinct real roots.
• When \( b^2 - 4ac = 0 \), there are exactly two equal real roots.
• When \( b^2 - 4ac < 0 \), the roots are complex.

In this exercise, the quadratic equation involves a complex number \( z \) and a condition placed on its real part, suggesting that these complex roots have specific properties linked to their real parts, such as being distinct.

Distinct Roots

The term "distinct roots" refers to the case where the roots of an equation are different from each other. For quadratic equations, this often relates to the discriminant \( b^2 - 4ac \). When the discriminant is not equal to zero, the roots are distinct.

In the context of complex numbers, two roots \( z_1 = 1 + yi \) and \( z_2 = 1 + xi \) are considered distinct if \( x eq y \). This implies that the imaginary parts of the two roots are different.

In the given exercise, the distinction in roots confirms that \( y eq 0 \), ensuring that the imaginary components of the roots are not zero. Since \( y^2 > 0 \), it impacts \( \beta = 1 + y^2 \), leading to the requirement that \( \beta > 1 \), allowing us to conclude \( \beta \in (1, \infty) \). Understanding this concept is essential in identifying the properties of the roots and selecting the correct answer choice.