Animated Solution for Mathematics - Complex Numbers: The complex numbers $z=x+iy$ which satisfy the equation $\left|\frac{z-5i}{z+5i}\right|=1$ lie on

Imagine you are standing in the middle of the complex plane, a vast, two-dimensional playground where every point is a complex number $z=x+iy$. Today, we are going to solve a problem that might look like a standard algebraic exercise, but is actually a beautiful geometric dance.

We are given the equation $\left|\frac{z-5i}{z+5i}\right|=1$ and asked to find the locus of $z$. At first glance, this fraction might seem intimidating, but let's break it down with the elegance of mathematical properties.

The first step in our journey is to simplify the expression. We have a powerful tool in our arsenal: the property that the modulus of a quotient is the quotient of the moduli, specifically $\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$.

Applying this to our equation, we get $\frac{|z-5i|}{|z+5i|}=1$. With a simple cross-multiplication, the fraction vanishes, leaving us with the much cleaner equation:

This is where the magic happens. In the complex plane, the expression $|z-z_0|$ is not just an algebraic term; it is the geometric distance between the point $z$ and the point $z_0$.

So, our equation is telling us something profound: the distance from $z$ to $5i$ must be exactly equal to the distance from $z$ to $-5i$.

Let's visualize this. We have two fixed points on the imaginary axis: $A=5i$ (which is the point $(0,5)$) and $B=-5i$ (which is the point $(0,-5)$).

The equation $|z-5i|=|z-(-5i)|$ tells us that our moving point $z$ must always maintain a perfect balance—it must be equidistant from $A$ and $B$. Think of it like a tug-of-war where the point $z$ is the rope, and it refuses to move closer to one point than the other.

From the fundamental principles of geometry, we know that the locus of all points equidistant from two fixed points is the perpendicular bisector of the line segment joining them.

Now, let's find that bisector. Our segment $AB$ connects $(0,5)$ and $(0,-5)$, which lies entirely on the y-axis.

The midpoint of this segment is:

This midpoint is the origin. Since the segment $AB$ is vertical, the line perpendicular to it must be horizontal.

A horizontal line passing through the origin is none other than the x-axis, or the real axis. Therefore, any complex number $z$ that satisfies our original equation must lie on the x-axis.

We have successfully navigated the algebra and arrived at a beautiful geometric conclusion. The locus is the x-axis. Keep this geometric intuition in your toolkit—it is the key to unlocking many such problems in your JEE journey!