Show that $|z|=1$ implies $|w|=1$

Question

Q:If $z$ and $w$ are two complex numbers connected by the equation $$w=\frac{2+3i-z}{1-(2-3i)z}$$ show that $|z|=1$ implies $|w|=1$.
i couldn't think of how to start.Any hints or solution will be appreciated.
Thanks in advance.

Answer 1

You may show it in the form

$$\bar w = \frac{1}{w}$$

So, set

• $a = 2+3i$ and note that
• $|z|=1 \Rightarrow \bar z = \frac{1}{z}$

$$\bar w = \overline{\frac{a-z}{1-\bar a z}} = \frac{\bar a - \frac{1}{z}}{1-a\frac{1}{z}}= \frac{z\bar a - 1}{z-a}=\frac{1- z\bar a}{a-z} = \frac{1}{w}$$

Answer 2

For any complex number $c$ we have $|c-z|^{2}= |1-\overset {-} c z|^{2}$ whenever $|z|=1$. (use the identity $|a+b|^{2}=|a|^{2}+|b|^{2}+2\Re a \overset {-} b$). Take $c=2+3i$. This gives $|w|^{2}=1$.