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Question Number 215417 by MATHEMATICSAM last updated on 06/Jan/25

a and b are complex numbers such that

∣ b ∣ = 1 . Find ∣ \frac{b - a}{1 - \overset{―}{a b}} ∣

Answered by MrGaster last updated on 06/Jan/25

Let a = x + y i, b = \cos θ + i \sin θ

b - a = (\cos θ - x) + i (\sin θ - y)

1 = \bar{a b} = 1 - (x + i y) (\cos θ - i \sin θ)

= 1 - (x \cos θ + y \sin θ) - i (x \sin θ - y \cos θ)

∣ b - a ∣= \sqrt{{(\cos θ - x)}^{2} + {(\sin θ - y)}^{2}}

∣ 1 - \bar{a b} ∣= \sqrt{{1 - x \cos θ - y \sin θ)}^{2} + {(x \sin θ - y \cos θ)}^{2}}

∣ \frac{b - a}{1 - \bar{a b}} ∣= \frac{\sqrt{{(\cos θ - x)}^{2} + {(\sin θ - y)}^{2}}}{\sqrt{{(1 - x \cos θ - y \sin θ)}^{2} + {(x \sin θ - y \cos θ)}^{2}}}

= 1

∣ \frac{b - a}{1 - \bar{a b}} ∣= 1 ✓

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