Animated Solution for Mathematics - Complex Numbers: All the points in the set $S=\left\{\frac{\alpha +i}{\alpha -i}:\alpha \in \mathbb{R}\right\}(i=\sqrt{-1})$ lie on a

My dear student, welcome to a fascinating journey into the heart of complex numbers. Often, when we see a complex expression like $z=\frac{\alpha +i}{\alpha -i}$, our first instinct is to dive into heavy algebra, rationalizing the denominator and separating real and imaginary parts.

While that works, it often obscures the elegant geometric truth hidden beneath the surface. Today, we are going to uncover that truth together.

When you encounter a complex fraction, especially one involving a real parameter $\alpha$, the modulus is your most powerful tool. The modulus operator has a beautiful property: the modulus of a quotient is the quotient of the moduli.

That is:

By taking the modulus of both sides of our equation, we transform a complex algebraic problem into a simple comparison of distances. Let us apply this to our expression:

Now, let us pause and visualize. In the complex plane, the number $\alpha +i$ represents a point with real part $\alpha$ and imaginary part $1$. The modulus $|\alpha +i|$ is the distance from the point $\alpha$ on the real axis to the point $i$ on the imaginary axis.

Similarly, the denominator $|\alpha -i|$ is the distance from the point $\alpha$ on the real axis to the point $-i$ on the imaginary axis. Because $\alpha$ is a real number, it lies perfectly on the horizontal axis.

This axis acts as a mirror, placing the point $i$ exactly one unit above it and the point $-i$ exactly one unit below it. Because of this symmetry, the distance from any point $\alpha$ on the real axis to $i$ must be identical to the distance from $\alpha$ to $-i$.

Let us confirm this with the cold, hard math. The modulus of $\alpha +i$ is $\sqrt{{\alpha }^2+1^2}=\sqrt{{\alpha }^2+1}$. The modulus of $\alpha -i$ is $\sqrt{{\alpha }^2+(-1{)}^2}=\sqrt{{\alpha }^2+1}$.

When we divide these two, we get:

The variables cancel out completely! This is the moment of magic. No matter what real value $\alpha$ takes, the modulus of $z$ is always $1$.

What does $|z|=1$ mean? It is the definition of a circle centered at the origin with a radius of $1$.

We have successfully navigated the algebra to find a beautiful geometric result. All points in the set $S$ lie on a circle whose radius is 1.

Keep this intuition in your toolkit: whenever you see a complex ratio, look for the modulus, look for the symmetry, and let the geometry guide you to the answer.