If α and β are different complex numbers with |β| = 1, then find |(^{β - α}/_{1 - ᾱβ})|

 

|(^{β - α}/_{1 - ᾱβ})|² = ^{|β - α|2}/_{|1 - ᾱβ|²}

= ^{(β - α)(β̄ - ᾱ)}/_{(1 - ᾱβ)(1 - αβ̄)}   [∵ |z|² = z z̄]

= ^{ββ̄ - βᾱ - αβ̄ + αᾱ}/_{1 - αβ̄ - ᾱβ + αᾱββ̄}

= ^{|β|2 - ᾱβ - αβ̄ + |α|2}/_{1 - αβ̄ - ᾱβ + |α|²|β|²}

Since |β| = 1 ⇒ |β|² = 1,

= ^{1 - ᾱβ - αβ̄ + |α|2}/_{1 - αβ̄ - ᾱβ + |α|²}

= 1

∴ |(^{β - α}/_{1 - ᾱβ})| = 1