TITLE: Proving that a complex number is real
QUESTION [20 upvotes]: If $$\left|\frac{Z_1 - iZ_2}{Z_1 + iZ_2}\right| = 1$$ then prove that $Z_1/Z_2$ is real . 

This is how I proceeded.
Dividing throughout by $Z_2$ we will have 
$$\left|{\frac{\frac{Z_1}{Z_2} - i}{\frac{Z_1}{Z_2} + i}}\right| = 1$$
Thus 
$$\left|\frac{Z_1}{Z_2} - i\right|= \left|\frac{Z_1}{Z_2} + i\right|$$
How do proceed from here ? 
According to the solution of the above problem , the previous statement would imply that $\frac{Z_1}{Z_2}$ is equidistant from $i$ and $-i$. Thus it is real. Now how does that make it real ? Please help me with this.

REPLY [9 votes]: Or (the "simpleton approach") you can let $\frac{Z_1}{Z_2} = x + yi$ where $x, y \in \mathbb R$
So you end up with $|x+(y+1)i| = |x+(y-1)i|$
which gives $x^2 + (y+1)^2 = x^2 + (y-1)^2 \implies (y+1)^2 - (y-1)^2 = 0 \implies 2y(2) = 0 \implies y = 0$. Since its imaginary part is zero, $\frac{Z_1}{Z_2}$ is real.