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Question
If α and β are different complex numbers with |β| = 1, then find `|(beta - alpha)/(1-baralphabeta)|`
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Solution
`|(beta - alpha)/(1 - baralpha beta)|^2`
= `((beta - alpha)/(1 - baralpha beta))bar(((beta - alpha)/(1 - baralpha beta))`
= `(beta - alpha)/(1 - baralpha beta) xx (barbeta - baralpha)/(1 - baralpha beta)`
= `(beta barbeta - baralphabeta - alpha barbeta + alpha baralpha)/(1 - alpha barbeta - baralphabeta +alphabaralpha.betabarbeta)`
= `(|beta|^2 - baralphabeta - alphabarbeta + |alpha|^2)/(1 - alphabarbeta - baralphabeta + |alpha|^2 . |beta|^2`)`
Given |β| = 1,
= `(1 + |alpha|^2 - baralphabeta - alphabarbeta)/(1 + |alpha|^2 - baralphabeta - alphabarbeta)`
= 1
∴ `|(beta - alpha)/(1 - baralphabeta)| = 1` or `|(beta - alpha)/(1 - baralphabeta)| = 1`
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