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If $A$ and $B$ are two different complex numbers and $|B| = 1$, find the value of $$\frac {|B-A| }{|1-\overline AB|}$$ where, as usual, $\overline A$ denotes the conjugate of $A$.

If possible please don't tell me the entire answer just tell me from where to begin.

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  • $\begingroup$ Hint: $B\times \overline B=1$. $\endgroup$ – lulu Sep 10 '17 at 10:54
  • $\begingroup$ Note: I reformatted your post, replacing your $A'$ with $\overline A$, the standard notation for complex conjugate. If you prefer it the way you wrote it, just click on "edit" and undo my change. $\endgroup$ – lulu Sep 10 '17 at 10:56
  • $\begingroup$ math.stackexchange.com/questions/506058/… $\endgroup$ – Guy Fsone Nov 23 '17 at 15:32
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Hint:)

$$\left(\frac {|B-A| }{|1-\overline AB|}\right)^2=\dfrac{B-A}{1-\overline AB}\dfrac{\overline B-\overline A}{1-A\overline B}$$ now multiply and simplify!

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Note that $|\overline{B}|=|B|=1$ and $$|1-\overline{A}B|=|1-\overline{A}B|\cdot|\overline{B}|=|(1-\overline{A}B)\cdot\overline{B}|=|\overline{B}-\overline{A}|B|^2|=|\overline{B}-\overline{A}|.$$ Can you take it from here?

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