If - - are different complex numbers with - -1- then - -1-

The correct option is A 1 We have, ββ̅= | β #160; |^2=1 ⇒β =1/β̅ Hence, #160; | α β/1 αβ | #160;= | α 1/β̅/1 αβ | #160;= | αβ ...

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Standard XII

Physics

Phase Difference

If α ,β are...

Question

If $α, β$ are different complex numbers with $| β | = 1$ , then $∣ ∣ ∣ \frac{¯ ¯¯ ¯ α - β}{1 - α β} ∣ ∣ ∣ =$

0

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Solution

The correct option is A $1$
We have,
$β ¯ β = {| β |}^{2} = 1$
$\Rightarrow β = \frac{1}{¯ β}$
Hence,
$∣ ∣ ∣ \frac{¯ ¯¯ ¯ α - β}{1 - α β} ∣ ∣ ∣$

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

$= ∣ ∣ ∣ \frac{¯ ¯¯ ¯ α ¯ β - 1}{1 - α β} ∣ ∣ ∣ \frac{1}{∣ ∣ ¯ β ∣ ∣}$
$= 1$ (since, $∣ ∣ ¯ α ¯ β - 1 ∣ ∣ = | 1 - α β |$ and $∣ ∣ ∣ \frac{1}{¯ β} ∣ ∣ ∣ = 1$ )

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