Problem 195 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(z\) and \(\omega\) are two non-zero complex numbers such that \(|z \omega|=1\), and \(\operatorname{Arg}(z)-\operatorname{Arg}(\omega)=\frac{\pi}{2}\), then \(\bar{Z} \omega\) is equal to (A) 1 (B) \(-1\) (C) \(i\) (D) \(-i\)

Step-by-Step Solution

Answer

(D) \(-i\)

Step 1: Understanding the Magnitude Condition

We are given that \(|z \omega| = 1\). This means that the product of the magnitudes of \(z\) and \(\omega\) is 1. Hence, \[|z| \times |\omega| = 1.\]

Step 2: Understanding the Argument Condition

The problem states \(\operatorname{Arg}(z)-\operatorname{Arg}(\omega)=\frac{\pi}{2}\). This implies the argument of \(z\) is 90 degrees greater than the argument of \(\omega\). Thus:\[\operatorname{Arg}(z \omega) = \operatorname{Arg}(z) + \operatorname{Arg}(\omega) = \frac{\pi}{2}.\]

Step 3: Analyze \(\bar{Z}\omega\)

Since \(\bar{Z}\omega = \frac{\omega}{z}\), we need to determine its value. Given\[\bar{z} = \frac{1}{z} \text{ because } |z|^2 = 1\, (|z||\omega| = 1).\]Thus, \[\bar{Z} \omega = \frac{\omega}{z}.\]

Step 4: Calculate \(\operatorname{Arg}(\bar{Z}\omega)\)

Since \(\operatorname{Arg}(\bar{z} \omega) = \operatorname{Arg}(\omega) - \operatorname{Arg}(z)\), and we know from Step 2 that \(\operatorname{Arg}(z)-\operatorname{Arg}(\omega)=\frac{\pi}{2}\), we have:\[-(\operatorname{Arg}(z) - \operatorname{Arg}(\omega)) = -\frac{\pi}{2}.\]

Step 5: Determine Final Value

Thus, the argument of \(\bar{Z} \omega\) is \[\operatorname{Arg}(\bar{Z} \omega) = -\frac{\pi}{2}.\]The complex number having an argument of \(-\pi/2\) is \(-i\). Thus,\[\bar{Z} \omega = -i.\]

Key Concepts

Magnitude of Complex Numbers

Understanding the magnitude of complex numbers is quite similar to understanding the length of the hypotenuse of a right triangle. Given a complex number, usually in the form of \( z = a + bi \), its magnitude is calculated using the formula:

• \( |z| = \sqrt{a^2 + b^2} \)

This formula derives from the Pythagorean theorem. The real part \(a\) and the imaginary part \(b\) form the two legs of the right triangle, making the magnitude of the complex number the hypotenuse.
In our exercise, we know for two complex numbers \(z\) and \(\omega\) that the product \(|z \, \omega| = 1\). Hence, it implies that the product of their individual magnitudes equals one, i.e.,

• \( |z| \times |\omega| = 1 \)

This relationship becomes particularly useful when solving problems involving these two complex numbers together.

Argument of Complex Numbers

The argument of a complex number helps us determine its angle in the complex plane. It is defined as the angle \( \theta \) that the line representing the complex number makes with the positive real axis, measured in the counter-clockwise direction. For a complex number \( z = a + bi \), the argument \( \operatorname{Arg}(z) \) is often given by:

• \( \operatorname{Arg}(z) = \tan^{-1}\left(\frac{b}{a}\right) \)

In the problem, the difference in the arguments of \(z\) and \(\omega\) is \( \frac{\pi}{2} \). This is significant because it tells us that \(z\) is rotated 90 degrees from \(\omega\).
Upon multiplication, the new argument \( \operatorname{Arg}(z \omega) \) results from summing their individual arguments:

• \( \operatorname{Arg}(z \omega) = \operatorname{Arg}(z) + \operatorname{Arg}(\omega) \)

This setup is crucial for solving many problems involving angles or the rotation of complex numbers.

Complex Conjugate

The complex conjugate of a complex number \( z = a + bi \) is \( \bar{z} = a - bi \). This means we simply switch the sign of the imaginary part. The complex conjugate has some interesting properties that are useful in complex number operations:

• The magnitude of a complex number and its conjugate are the same: \( |z| = |\bar{z}| \).
• If \( z \) is multiplied by its conjugate, we end up with a real number: \( z \bar{z} = a^2 + b^2 = |z|^2 \).

In the exercise, \( \bar{Z} \omega \) is investigated, leading us to the identity:

• \( \bar{Z} = \frac{1}{z} \text{ if } |z|^2 = 1. \)

Thus, complex conjugates not only help in simplifying expressions but also play a role in dividing and interpreting complex numbers within unit magnitude conditions.

Complex Multiplication

Complex multiplication involves multiplying two complex numbers together, resulting in a new complex number. Given two complex numbers \( z_1 = a + bi \) and \( z_2 = c + di \), the product \( z_1 z_2 \) is calculated as:

• \( (a + bi)(c + di) = (ac - bd) + (ad + bc)i \)

This operation combines both multiplication and addition of real and imaginary components.
Furthermore, when multiplying complex numbers, their magnitudes are also multiplied and their arguments (or angles) are added:

• \( |z_1 z_2| = |z_1| \times |z_2| \)
• \( \operatorname{Arg}(z_1 z_2) = \operatorname{Arg}(z_1) + \operatorname{Arg}(z_2) \)

In our exercise, understanding the multiplication and the resulting arguments of complex numbers is pivotal, particularly using the identity \( \bar{Z}\omega = \frac{\omega}{z} \), which directly leverages the multiplication properties to attain the problem's solution.