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As the question says, I have to find $\big|\frac{\beta-\alpha}{1-\bar{\alpha}\beta}\big|$ given that $|\beta| = 1$. Where $\bar{\alpha}$ refers to the complex conjugate of $\alpha$.

Unfortunately, I don't see how to do this. If I write $\alpha = a + ib$ and $\beta = c + id$ and try to evaluate the expression, I end up with a large hairy mess which I can't simplify, not to mention the fact that it doesn't make use of the information $|\beta| = 1$.

Note: Please provide a hint on how to proceed, not the complete answer.

Lee Mosher
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Aayush Agrawal
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  • Almost surely related: http://math.stackexchange.com/questions/343982/prove-if-z-1-and-w-1-then-1-zw-neq-0-and-z-w-over-1. – Martín-Blas Pérez Pinilla Feb 14 '16 at 18:19
  • @Martín-BlasPérezPinilla I don't think so. For one, the assumption in the above question is that both the complex numbers are less than zero. Second, this question is asking for a hint, and not the complete solution. Edit: I read your comment wrong. Although i can't read the above question's solutions as i still want to try and solve this by myself. – Aayush Agrawal Feb 14 '16 at 18:20
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    Hint: $|\beta| = 1 \iff \beta \overline\beta=1,$, so the denominator $1-\overline{\alpha}\beta = \beta \overline\beta - \overline{\alpha}\beta = \beta,\overline{(\beta-\alpha)}$. – dxiv Apr 29 '18 at 17:39

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