Problem 171 - A Complete Resource Book in Mathematics for JEE Main

Question

One of the values of \((a+i b)^{m i n}+(a-i b)^{m / n}\) is (A) \(2\left(a^{2}+b^{2}\right)^{m / n} \cos \left(\frac{m}{n} \tan ^{-1} \frac{b}{a}\right)\) (B) \(2\left(a^{2}+b^{2}\right)^{m / 2 n} \cos \left(\frac{m}{n} \tan ^{-1} \frac{b}{a}\right)\)

Step-by-Step Solution

Answer

The correct answer is (B).

Step 1: Express the Expressions in Polar Form

Given the complex numbers \((a + ib)\) and \((a - ib)\), observe that both can be expressed in their polar forms. Using the identity for complex numbers in polar form, \(z = r(\cos \theta + i\sin \theta)\), we rewrite \(a + ib\) as \(\sqrt{a^2 + b^2}(\cos \theta + i\sin \theta)\), where \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). Similarly, \(a - ib = \sqrt{a^2 + b^2}(\cos \theta - i\sin \theta)\).

Step 2: Apply De Moivre's Theorem

Using De Moivre's Theorem, which is \((r(\cos \theta + i\sin \theta))^n = r^n (\cos(n \theta) + i\sin(n \theta))\), compute \((a+ib)^{m/n}\) and \((a-ib)^{m/n}\). Hence, \((a + ib)^{m/n} = (a^2 + b^2)^{m/(2n)} (\cos(\frac{m}{n}\theta) + i\sin(\frac{m}{n}\theta))\) and \((a - ib)^{m/n} = (a^2 + b^2)^{m/(2n)} (\cos(\frac{m}{n}\theta) - i\sin(\frac{m}{n}\theta))\).

Step 3: Sum the Two Expressions

Adding the expressions from Step 2, note that the imaginary parts cancel out because \(\sin\) terms have opposite signs: \[(a + ib)^{m/n} + (a - ib)^{m/n} = 2(a^2 + b^2)^{m/(2n)} \cos(\frac{m}{n}\theta)\].

Step 4: Simplify and Compare with Options

The simplified expression for \((a + ib)^{m/n} + (a - ib)^{m/n}\) is accurately given by: \[2(a^2 + b^2)^{m/(2n)} \cos\left(\frac{m}{n} \tan^{-1}\frac{b}{a}\right)\]. This corresponds to option (B).

Key Concepts

Polar Form

Complex numbers, like the ones you see with real and imaginary parts such as \(a + ib\), can be expressed using polar coordinates. This is known as the polar form. Polar form leverages the magnitude and direction of a complex number, rather than expressing it directly with real and imaginary parts.

To convert a complex number \(z = a + ib\) into polar form, you will need to determine two things:

• The magnitude \(r = \sqrt{a^2 + b^2}\), which represents the distance of \(z\) from the origin in the complex plane.
• The angle \(\theta = \tan^{-1}(b/a)\), called the argument, which is the angle \(z\) makes with the positive real axis.

Thus, a complex number \(z\) in polar form is represented as \(z = r (\cos \theta + i \sin \theta)\). This representation is incredibly useful when you need to perform operations like multiplication and division of complex numbers.

In mathematical contexts, expressing complex numbers in polar form simplifies the application of De Moivre's Theorem, among other things.

De Moivre's Theorem

De Moivre's Theorem provides a powerful formula for raising complex numbers in polar form to any integer power. The formula is expressed as \([r(\cos \theta + i\sin \theta)]^n = r^n (\cos(n\theta) + i \sin(n\theta))\), where \(n\) is an integer.

This theorem makes it extremely convenient to work with powers of complex numbers because it allows you to easily compute the power of the magnitude and multiply the angle. For example, if you have a complex number \(z\) in polar form and want to calculate \(z^{m/n}\), you can rewrite \(z\) with De Moivre's method to simplify the exponentiation process:

• Raise the magnitude to the power specified.
• Multiply the argument (angle \(\theta\)) by the power \(n\).

Using this theorem, what's traditionally a complicated operation turns into simple multiplication and exponentiation. It is especially useful for simplifying expressions involving powers and roots of complex numbers, making it a staple tool in complex number algebra.

Imaginary Unit

In the world of complex numbers, the imaginary unit \(i\) plays a central role. It is defined as \(i = \sqrt{-1}\), a number that does not exist on the real number line. However, the introduction of \(i\) makes it possible to solve equations like \(x^2 = -1\), which have no real solutions.

When we write a complex number \(a + ib\), \(i\) denotes that the "\(b\)" part of the number exists along the imaginary axis of the complex plane, perpendicular to the real axis.

• \(i^2 = -1\) is the foundational property of \(i\).
• \(i^3 = -i\) and \(i^4 = 1\) follow as subsequent powers.

These properties of \(i\) enable algebra with complex numbers, including addition, subtraction, multiplication, division, and even raising to powers or finding roots. Understanding \(i\) is essential to work fluently with complex numbers, enabling a deeper grasp of mathematical subjects such as electromagnetic theory, quantum mechanics, and signal processing.