Animated Solution for Mathematics - Complex Numbers: If $z$ and $w$ are two complex numbers such that $|zw|=1$ and $\arg (z)-\arg (w)=\pi /2$, then :

Welcome, future engineers and mathematicians! Today, we are going to unravel a problem that might look like a simple algebraic manipulation at first glance, but it is actually a beautiful exploration of the geometry of the Argand plane.

When we deal with complex numbers $z$ and $w$, we are not just dealing with abstract variables; we are dealing with vectors that have both a length and a direction. Let's break this down step-by-step.

The Polar Transformation

To truly master complex numbers, we must embrace Euler's polar form. It is the most powerful tool in our arsenal.

Let's represent our complex numbers as:

Here, $r_1$ and $r_2$ are the magnitudes (or moduli) of the vectors, and ${\theta }_1$ and ${\theta }_2$ are their respective arguments. By defining them this way, we transform the daunting task of multiplication into a simple operation of exponents.

Whenever you see modulus and argument conditions in a JEE problem, think polar form immediately!

Decoding the Clues

The problem provides us with two vital pieces of information. First, we are told that $|zw|=1$.

Using the property that the modulus of a product is the product of the moduli, we get $|z|\cdot |w|=1$, which translates to $r_1{r}_2=1$. This is our first anchor.

Second, we are given $\arg (z)-\arg (w)=\frac{\pi }{2}$. In our polar representation, this is simply ${\theta }_1-{\theta }_2=\frac{\pi }{2}$.

Geometrically, this tells us that $z$ is essentially $w$ rotated by $90^{\circ }$ counter-clockwise. It is a precise geometric relationship that we will now use to solve the expression.

The Conjugate Dance

Now, let's look at the expressions in the options. They involve the conjugate $\bar{z}$. Remember, taking the conjugate is a reflection across the real axis.

If $z=r_1{e}^{i{\theta }_1}$, then its conjugate is $\bar{z}=r_1{e}^{-i{\theta }_1}$. The magnitude remains unchanged, but the angle is negated.

When we multiply $\bar{z}$ by $w$, we get:

Using the laws of exponents, this becomes:

The Final Synthesis

We are almost there! We know $r_1{r}_2=1$. We also know that ${\theta }_1-{\theta }_2=\frac{\pi }{2}$, which implies that ${\theta }_2-{\theta }_1=-\frac{\pi }{2}$.

Substituting these into our expression, we get:

Using Euler's identity, $e^{-i\frac{\pi }{2}}=\cos (-\frac{\pi }{2})+i\sin (-\frac{\pi }{2})=0+i(-1)=-i$.

Similarly, for $z\bar{w}$, we would find:

By systematically applying these properties, we have navigated the complexity and arrived at the core truth. Keep practicing this visualization, and you will find that complex numbers become your favorite part of the JEE syllabus!