Problem 65 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(\alpha, \beta\) be real and \(z\) be 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) \(\beta \in(1, \infty)\) (B) \(\beta \in(0,1)\) (C) \(\beta \in(-1,0)\) (D) \(|\beta|=1\)

Step-by-Step Solution

Answer

Option (A) is correct: \(\beta \in(1, \infty)\).

Step 1: Understand the condition for roots

The condition for two distinct roots of the quadratic equation is that the two roots must have the form \(z_1 = 1 + i\gamma\) and \(z_2 = 1 - i\gamma\), where \(\gamma\) is a non-zero real number, because they lie on the line where \(\operatorname{Re} z = 1\).

Step 2: Roots Representation

Express the roots \(z_1 = 1 + i\gamma\) and \(z_2 = 1 - i\gamma\) in the form \(z = x + iy\), then substitute these into the expression \(z^2 + \alpha z + \beta = 0\).

Step 3: Vieta's Formulas Application

According to Vieta's formulas, for a quadratic equation \(az^2 + bz + c = 0\) with roots \(z_1\) and \(z_2\), \(z_1 + z_2 = -\frac{b}{a}\) and \(z_1 z_2 = \frac{c}{a}\). For \(z^2 + \alpha z + \beta = 0\), this means \(z_1 + z_2 = -\alpha = 2\) and \(z_1 z_2 = \beta = (1 + i\gamma)(1 - i\gamma)\).

Step 4: Calculate the Product of Roots

Compute \((1 + i\gamma)(1 - i\gamma) = 1 - (i\gamma)^2 = 1 - (-\gamma^2) = 1 + \gamma^2\). Thus, \(\beta = 1 + \gamma^2\), using the property of squares \(i^2 = -1\).

Step 5: Determine Conditions for \(\beta\)

Since \(\gamma\) is non-zero for the roots to be distinct, \(\gamma^2 > 0\). Therefore, \(\beta = 1 + \gamma^2 > 1\).

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers, consisting of two parts: a real part and an imaginary part. They are written in the form \( z = x + iy \), where \( x \) is the real part, and \( y \) is the imaginary part. The imaginary unit \( i \) satisfies \( i^2 = -1 \). So, every complex number can be visualized as a point in the complex plane with an x-coordinate of \( x \) and a y-coordinate of \( y \), often called the Argand diagram.
Complex numbers become particularly useful when working with quadratic equations with complex roots. They allow for a complete domain to solve any quadratic equation, with their roots being either real or complex.

• A key property of complex numbers is that they are closed under addition, subtraction, multiplication, and division (except by zero).
• Conjugate pairs often appear as roots of quadratic equations when the coefficients are real numbers.
• In our problem, \( z_1 = 1 + i\gamma \) and \( z_2 = 1 - i\gamma \) represent complex conjugates.

Vieta's Formulas

Vieta's formulas provide a direct relationship between the coefficients of a polynomial and the roots of the polynomial. For a quadratic equation of the form \( az^2 + bz + c = 0 \), Vieta's formulas state:

• The sum of the roots \( z_1 + z_2 \) is equal to \(-\frac{b}{a}\).
• The product of the roots \( z_1 z_2 \) is equal to \( \frac{c}{a} \).

This relationship holds true regardless of whether the roots are real or complex. In the context of the exercise, the equation \( z^2 + \alpha z + \beta = 0 \) has roots \( z_1 \) and \( z_2 \). Here, using Vieta's formulas helps simplify the problem:

• We find \( z_1 + z_2 = -\alpha \). Given that the roots lie on \( \operatorname{Re} z = 1 \), this gives us \(-\alpha = 2\), leading to \( \alpha = -2 \).
• The product \( z_1 z_2 = \beta \) gives \( \beta = 1 + \gamma^2 \). This implies \( \beta \) depends on \( \gamma \), showing its nature based on the quadratic characteristics of the roots.

Roots of Equations

The roots of a quadratic equation are the solutions that satisfy the equation \( az^2 + bz + c = 0 \). The nature of the roots - whether they are real or complex - is determined by the discriminant \( \Delta = b^2 - 4ac \).

• If \( \Delta > 0 \), the equation has two distinct real roots.
• If \( \Delta = 0 \), the equation has a repeated real root.
• If \( \Delta < 0 \), the equation has two distinct complex roots, which are conjugates of each other.

In the given exercise, since it discusses roots on the line \( \operatorname{Re} z = 1 \), we are considering complex roots of the form \( 1 \pm i\gamma \). These roots are important as they satisfy the condition of having a real part of 1, while still being distinct, due to non-zero \( \gamma \).
For the quadratic \( z^2 + \alpha z + \beta = 0 \), determining \( \beta = 1 + \gamma^2 \) stems from recognizing the specific form of these complex roots, as their product involves the square of the imaginary component \( \gamma \). This relationship is essential for concluding the range of \( \beta \), ultimately finding that \( \beta > 1 \), based on \( \gamma^2 > 0 \) ensuring the roots' distinction.