Browse · MATH

jmc

algebra intermediate

Problem

Let

α

and

β

be complex numbers such that

∣ β ∣ = 1

and

\overline{α} β \neq = 1.

Find the maximum value of

\frac{β - α}{1 - α β} .

Solution

Since

∣ β ∣ = 1,

∣ \overline{β} ∣ = 1,

so

\frac{β - α}{1 - α β} = \frac{1}{∣ β ∣} \cdot \frac{β - α}{1 - α β} = \frac{β - α}{β - α β β} = \frac{β - α}{β - α ∣ β ∣ ^{2}} = \frac{β - α}{β - α} = 1 .

Final answer

1