Animated Solution for Mathematics - Complex Numbers: Let $z$ and $\omega$ be two non zero complex numbers such that $|z|=|\omega |$ and $Argz+Arg\omega =\pi$, then $z$ equals

Welcome, fellow traveler on the path to JEE mastery! Today, we are not just solving an algebraic equation; we are embarking on a journey through the Argand plane.

Complex numbers are often misunderstood as mere abstract entities, but they are, in fact, the language of rotation and reflection. When we look at a complex number $\omega$, we aren't just looking at a point; we are looking at a vector with a magnitude $r$ and an angle $\theta$.

Let us place $\omega$ on the stage. We define it in its most elegant form, the polar form:

This is the heartbeat of complex number theory. It tells us exactly where $\omega$ lives: at a distance $r$ from the origin, rotated by an angle $\theta$ from the positive real axis. Keep this image in your mind; it is the anchor for everything we are about to do.

Now, consider our second actor, $z$. The problem gives us two crucial clues: $|z|=|\omega |=r$ and $Arg(z)+Arg(\omega )=\pi$.

If we let $Arg(\omega )=\theta$, then the argument of $z$ is locked in at $Arg(z)=\pi -\theta$. This is the moment where many students stumble.

But look at the geometry! An angle of $\pi -\theta$ is simply the reflection of $\theta$ across the imaginary axis. It is a perfect, symmetric counterpart.

So, we write $z$ in its polar form:

This is where we must be precise. We need to evaluate $\cos (\pi -\theta )$ and $\sin (\pi -\theta )$. Recall your unit circle.

In the second quadrant, the cosine is negative, and the sine is positive. Thus, we have the identities:

Substituting these back into our expression for $z$, we get:

Now, let us look at the complex conjugate of $\omega$, which is $\bar{\omega }$. Geometrically, $\bar{\omega }$ is the reflection of $\omega$ across the real axis. Its polar form is $\bar{\omega }=r(\cos \theta -i\sin \theta )$.

Look closely at our expression for $z$ and the expression for $\bar{\omega }$. They are almost identical, but the signs are flipped. If we factor out a negative sign from our expression for $z$, we get:

And there it is! The term in the parentheses is exactly $\bar{\omega }$. Therefore, we arrive at the beautiful, elegant conclusion:

This result is not just a collection of symbols; it is a geometric truth. We found that $z$ is the reflection of $\bar{\omega }$ through the origin.

By visualizing the Argand plane and respecting the trigonometric identities, we turned a potentially confusing problem into a clear, logical path. Keep this mindset—visualize, simplify, and trust the math—and you will conquer any problem the JEE throws your way!