If - and - are complex numbers with -1- find -1-

|β| = 1 ⇒ |β|^2 = 1 ⇒ββ = 1 β α/1 αβ = β α/ββ αβ = β α/β (β α) = 1/β β α/β α = 1 ×β α/(β α) = 1 ...

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Standard XII

Mathematics

Dot Product of Two Vectors

If α and β ar...

Question

If $α$ and $β$ are complex numbers with $| β | = 1$ , find $∣ ∣ \begin{matrix} \frac{β - α}{1 - ¯ ¯ ¯ α β} \end{matrix} ∣ ∣$ .

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Solution

$| β | = 1 \Rightarrow | β |^{2} = 1 \Rightarrow β ¯ ¯ ¯ β = 1$

$∣ ∣ \begin{matrix} \frac{β - α}{1 - ¯ ¯ ¯ α β} \end{matrix} ∣ ∣ = ∣ ∣ ∣ \begin{matrix} \frac{β - α}{β ¯ ¯ ¯ β - ¯ ¯ ¯ α β} \end{matrix} ∣ ∣ ∣$

= $∣ ∣ ∣ \begin{matrix} \frac{β - α}{β (¯ ¯ ¯ β - ¯ ¯ ¯ α)} \end{matrix} ∣ ∣ ∣ = ∣ ∣ \begin{matrix} \frac{1}{β} \end{matrix} ∣ ∣ ∣ ∣ ∣ \begin{matrix} \frac{β - α}{¯ ¯ ¯ β - ¯ ¯ ¯ α} \end{matrix} ∣ ∣ ∣$

= $1 \times ∣ ∣ ∣ \begin{matrix} \frac{β - α}{(¯ ¯¯¯¯¯¯¯ ¯ β - α)} \end{matrix} ∣ ∣ ∣$

= 1

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If α and β are complex numbers with |β|=1, find ∣∣β−α1−¯¯¯αβ∣∣.

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