How to solve $\left|\frac{1 + a + bi}{1 + b - ai}\right| = 1$ I have a problem with solving following equation:
$$\left|\frac{1 + a + bi}{1 + b - ai}\right| = 1$$ (where $a$, $b$ are real numbers and $i$ is an imaginary unit) 
I tried to simplify its left side to something like $c + di$ but I don't know any method to achieve it in this case. Do you have any ideas how do it?

You may rearrange:
\begin{align*}
\vert a + b \mathrm{i} + 1 \vert = \vert b - a \mathrm{i} +1 \vert \\
\end{align*}
Let's introduce $z = a+b \mathrm{i}$, the equation becomes
\begin{align*}
&\vert z + 1 \vert = \vert -\mathrm{i} z +1 \vert \\
&\vert z + 1 \vert = \vert - \mathrm{i} \vert \vert z + \mathrm{i} \vert \\
&\vert z + 1 \vert = \vert z + \mathrm{i} \vert \\
\end{align*}
The equation in words: "find all complex numbers $z$ for which the distance in the complex plane to the point $-1$ is equal to the distance to the point $-\mathrm{i}$".
The solution is all points $z$ that lie on the line through the origin of the complex plane with slope $1$, i.e. all $z = a + b \mathrm{i}$ for which $a = b$.
