# If alpha and beta are different complex number whit modulus of beta = 1, then find the modulus of ((beta - alpha)(1)-( conjugate of alpha)(beta))) I have to find modulus of whole question (means (β-α)/(1-(conjugate of α)β).

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Luca B. | Certified Educator

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You need to consider the complex numbers ` alpha` and `beta,` such that:

`alpha = a + b*i`

`beta = c + d*i`

The problem provides the information that the absolute value of the complex number `beta` is 1, such that:

`|beta| = sqrt(c^2 + d^2) = 1 => c^2 + d^2 = 1 => c = 0 , d = 1 or c = 1, d = 0.`

You need to evaluate `|(beta - alpha)/(1 - bar alpha*beta)|` , such that:

`|(beta - alpha)/(1 - bar alpha*beta)| = |(beta - alpha)|/|(1 - bar alpha*beta)|`

You need to evaluate beta - alpha, such that:

`beta - alpha = (a - c) + (b - d)*i => |(beta - alpha)| = sqrt((a - c)^2 + (b - d)^2)`

You need to evaluate 1 - bar alpha*beta, such that:

`1 - (a - b*i)(c + d*i) = 1 - (ac + bd + i*(ad - bc))`

`1 - (a - b*i)(c + d*i) = 1 - (ac + bd) - i*(ad - bc)`

`|1 - (a - b*i)(c + d*i)| = sqrt((1 - (ac + bd))^2 + (ad - bc)^2)`

`|(beta - alpha)|/|(1 - bar alpha*beta)| = (sqrt((a - c)^2 + (b - d)^2))/(sqrt((1 - (ac + bd))^2 + (ad - bc)^2))`

Considering `c = 0 , d = 1` , yields:

`|(beta - alpha)|/|(1 - bar alpha*beta)| = (sqrt(a^2 + (b - 1)^2))/(sqrt((1 - b)^2 + a^2)) = 1`

Considering `d = 0 , c = 1` , yields:

`|(beta - alpha)|/|(1 - bar alpha*beta)| = (sqrt((a - 1)^2 + b^2))/(sqrt((1 - a)^2 + (b)^2)) = 1`

Hence, evaluating the absolute value of the complex number `|(beta - alpha)|/|(1 - bar alpha*beta)|` , under the given conditions, yields `|(beta - alpha)|/|(1 - bar alpha*beta)| = 1.`

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