First slide

Algebra of complex numbers

Question

If $α, β$ are distinct complex numbers with $|β| = 1,$ then value of $|\frac{β - α}{1 - \bar{α} β}|$ equals

Moderate

A

1
B

$|α|$
C

2
D

none of these

Solution

$|β| = 1 \Rightarrow β \bar{β} = 1$
$\begin{array}{l} ∴ |\frac{β - α}{1 - \bar{α} β}| = |\frac{β - α}{1 - \bar{α} / \bar{β}}| = |\frac{\bar{β} (β - α)}{\bar{β} - \bar{α}}| = | β | \frac{| β - α |}{| \bar{β} - \bar{α} |} \\ = | β | (1) = 1 . [∵ | z | = | \bar{z} |] \end{array}$

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