Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 1A / b

Table of contents
- Solution b

Solution

Write

z = x + i y

. Rationalizing gives

\frac{z - 2}{z} = \frac{x ^{2} - 2 x + y ^{2} + 2 i y}{x ^{2} + y ^{2}} .

An argument of

π /4

requires equal positive real and imaginary parts. Hence

x^{2} - 2 x + y^{2} = 2 y, y > 0.

Completing squares yields

(x - 1)^{2} + (y - 1)^{2} = 2, y > 0 .

Thus the constant-argument locus of a Möbius transformation is the part above the real axis of the circle with centre

(1, 1)

2

. Its endpoints

z = 0

and

z = 2

are excluded because the quotient is respectively undefined and zero.

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