Animated Solution for Mathematics - Complex Numbers: If $\omega =\frac{z}{z-\frac{1}{3}i}$ and $|\omega |=1$, then $z$ lies on

Imagine you are standing on the Argand plane, a vast, two-dimensional canvas where complex numbers live. You are given a mysterious relationship:

with the constraint that the modulus of $\omega$ is exactly one, $|\omega |=1$. Your mission is to uncover the path, or the locus, that the complex number $z$ traces as it dances across this plane.

Our First Key

We start with the condition $|\omega |=1$. Substituting our expression for $\omega$, we get:

Here, we invoke a powerful property of complex numbers: the modulus of a quotient is the quotient of the moduli, $\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$. Applying this, our equation transforms into:

By multiplying both sides by the denominator, we arrive at the beautiful, symmetric relation:

Now, pause and look at this equation. $|z|$ represents the distance of the point $z$ from the origin $(0,0)$, while $|z-\frac{1}{3}i|$ represents the distance of the point $z$ from the fixed point $\frac{1}{3}i$ (or $(0,\frac{1}{3})$ in Cartesian coordinates).

The equation is telling us something profound: the point $z$ is always at the same distance from the origin as it is from the point $\frac{1}{3}i$. If you walk such that you are always equidistant from these two stakes, you are walking along the perpendicular bisector of the line segment connecting them.

Since the perpendicular bisector of any line segment is, by definition, a straight line, we have our answer. The locus of $z$ is a straight line.

Specifically, it is the horizontal line $y=\frac{1}{6}$, which sits exactly halfway between $y=0$ and $y=\frac{1}{3}$. We have navigated from a complex algebraic fraction to a simple, elegant geometric truth. This is the beauty of complex numbers—they are not just numbers; they are the language of geometry itself.