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Properties of Modulus

let z1 and ...

Question

let $z_{1}$ and $z_{2}$ be two complex number such that $| 1 - z_{1} z_{2} |^{2} - | z_{1} - z_{2} |^{2} = k (1 - | z_{1} |^{2}) (1 - | z_{2} |^{2})$ find the value of $k$

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Solution

Given,

$| 1 - {¯ z}_{1} z_{2} |^{2} - | z_{1} - z_{2} |^{2}$

$= (1 + z_{1}^{2} z_{2}^{2} - 2 z_{1} z_{2}) - (z_{1}^{2} + z_{2}^{2} - 2 z_{1} z_{2})$

$= 1 + z_{1}^{2} z_{2}^{2} - 2 z_{1} z_{2} - z_{1}^{2} - z_{2}^{2} + 2 z_{1} z_{2}$

$= 1 + z_{1}^{2} z_{2}^{2} - z_{1}^{2} - z_{2}^{2}$ .......(1)

$(1 - | z_{1} |^{2}) (1 - | z_{2} |^{2})$

$= 1 - | z_{2} |^{2} - | z_{1} |^{2} + | z_{1} |^{2} | z_{2} |^{2}$ .......(2)

Comparing (1) and (2), it's clear that,

$| 1 - z_{1} {¯ z}_{2} |^{2} - | z_{1} - z_{2} |^{2} = (1 - | z_{1} |^{2}) (1 - | z_{2} |^{2})$

$∴ k = 1$

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be two complex numbers such that

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