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

Question

If \(\alpha, \beta\) are the roots of \(z+\frac{1}{z}=2(\cos \theta+\sin \theta)\) Then, (A) \(|\alpha-i|>|\beta-i|\) (B) \(|\alpha-i|<|\beta-i|\) (C) \(|\alpha-i|=|i-\beta|\) (D) \(|\alpha-i|=|\beta-i|\)

Step-by-Step Solution

Answer

Option (D): \(|\alpha-i|=|\beta-i|\).

Step 1: Understanding the Equation

The equation given is \( z + \frac{1}{z}=2(\cos \theta + \sin \theta) \). This can be interpreted in the complex plane where \( z = r e^{i\theta} \) is a complex number with roots \( \alpha \) and \( \beta \), and \( |z| = 1 \) since \( z \cdot \frac{1}{z} = 1 \).

Step 2: Rewrite the Equation for Roots

Rewrite the equation as a quadratic: \( z^2 - 2(\cos \theta + \sin \theta)z + 1 = 0 \).

Step 3: Determine Roots using Vieta's Formulas

Vieta's formulas tell us the roots \( \alpha \) and \( \beta \) are given by: \( \alpha + \beta = 2(\cos \theta + \sin \theta) \) and \( \alpha\beta = 1 \).

Step 4: Substitute and Solve for Roots

Given \( \alpha + \beta = 2(\cos \theta + \sin \theta) \), write roots as \( \alpha = e^{i(\theta + \phi)} \) and \( \beta = e^{i(\theta - \phi)} \), based on their angular symmetry.

Step 5: Calculate Distances and Compare

Because the magnitude of both roots \( |\alpha| = |\beta| = 1 \) and they lie symmetrically about the angle \( \theta \), the distances \( |\alpha - i| \) and \( |\beta - i| \) are equal. Thus, \[ |\alpha - i| = |\beta - i|. \]

Step 6: Conclusion

Therefore, the correct answer is that the distances from the imaginary axis are equal for both roots. Thus, \(|\alpha - i| = |\beta - i|\), which corresponds to option (D).

Key Concepts

Vieta's Formulas

Vieta's Formulas are a fantastic way to link the coefficients of a polynomial to the sum and product of its roots. They're especially handy when solving quadratic equations. For any quadratic equation of the form:

• \( ax^2 + bx + c = 0 \)

The formulas are:

• The sum of the roots, \( \alpha + \beta = -\frac{b}{a} \)
• The product of the roots, \( \alpha\beta = \frac{c}{a} \)

In the context of our problem, the equation was rewritten as a quadratic:

• \( z^2 - 2(\cos \theta + \sin \theta)z + 1 = 0 \)

By applying Vieta's Formulas, we find:

• The sum of the roots \( \alpha + \beta = 2(\cos \theta + \sin \theta) \)
• The product of the roots \( \alpha\beta = 1 \)

These relationships help us understand more about the nature of the roots \( \alpha \) and \( \beta \). They're part of what makes solving quadratic equations not just possible, but sometimes even elegant.

Quadratic Equations

Quadratic equations are at the heart of algebra. They usually appear in the form:

• \( ax^2 + bx + c = 0 \)

These equations have degree 2, implying that they typically have two solutions or roots. The process of finding these roots can be done using various methods like factoring, completing the square, or applying the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In our mathematical journey, quadratic equations are rewritten to simplify the problem. In the example problem, the equation \( z + \frac{1}{z} = 2(\cos \theta + \sin \theta) \) is transformed into a quadratic:

• \( z^2 - 2(\cos \theta + \sin \theta)z + 1 = 0 \)

This transformation is crucial because it allows us to use tools like Vieta's formulas to easily identify properties of roots \( \alpha \) and \( \beta \). This makes solving the equation more straightforward, revealing insightful geometric interpretations on the complex plane.

Complex Plane

The complex plane is a two-dimensional plane where we can visualize complex numbers. Each complex number is represented as a point or a vector in this plane, defined by:

• \( z = x + yi \)

Where \( x \) is the real part, and \( y \) the imaginary part. The complex plane is especially useful when dealing with equations involving complex numbers. It lets us explore both algebraic and geometric aspects of these numbers. In our exercise, the complex number root \( z \) is expressed in the polar form \( z = re^{i\theta} \), where:

• \( |z| = r \) is the magnitude
• \( \theta \) is the angle formed with the positive real axis

A key part of solving the exercise involved understanding that \(|z| = 1\), positioning our roots geometrically on the unit circle in the complex plane. This offers insight into their symmetrical nature around angle \( \theta \). By comparing the magnitudes of \( \alpha \) and \( \beta \), we determine that their distances from any point (like \( i \)) are equal, highlighting the beauty of symmetry and distance in complex analysis.