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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?

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  • $\begingroup$ it is equivalent to $\frac{\sqrt{(1+a)^2+b^2}}{\sqrt{(1+b)^2+a^2}}=1$ $\endgroup$ – Dr. Sonnhard Graubner Oct 6 '14 at 19:09
  • $\begingroup$ That helped, thanks a lot :) $\endgroup$ – Michał Oct 6 '14 at 19:24
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Hint:

Use the fact that for any complex number $z_1$ and $z_2$, $$\left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}$$ Then try and rearrange.

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  • $\begingroup$ That helped, thanks a lot :) $\endgroup$ – Michał Oct 6 '14 at 19:25
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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 $a = b$.

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Write $z=a+bi$, so your equation be written as $$\left|\frac{1+z}{1-iz}\right| = 1$$ (verify this!). This is equivalent to $$|1+z| = |1-iz|$$ as long as we remember that $z\neq -i$ (to ensure a nonzero denominator in the original equation). The RHS can be written as $|-i(i+z)|$, and since $|-i|=1$, we can write the equation as $$|z-(-1)| = |z-(-i)|$$

This equation says that the solutions are the points in $\mathbb C$ that are equidistant from $-1$ and $-i$. These are precisely the points on the perpendicular bisector of the segment joining $-1$ and $-i$, which is the diagonal line through the origin, so the solutions are the points of the form $$\boxed{z = t + ti}$$ for real $t$. Note that $-i$ is not among them (if it were, we would have to exclude it because it would make the LHS of the original equation undefined).

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