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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/11/c/iv/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 11 / c / iv / Solution by

Codex  0 2026-10-18

For the reciprocal Möbius transformation, substitute

z = 1/ w

,

\overset{z}{ˉ} = 1/ \overset{w}{ˉ}

and multiply by

w \overset{w}{ˉ}

, which is nonzero on the stipulated domain. This yields

a + b w + \overset{ˉ}{b} \overset{w}{ˉ} + c w \overset{w}{ˉ} = 0.

Consequently

A = c, B = \overset{ˉ}{b}, C = a, A = 0 ⟺ c = 0 .

For a nondegenerate original circle,

c = 0

says that it passes through the origin, and its image is a straight line. The restriction

w \neq = 0

must be retained: multiplying through does not authorize adding that excluded point. Degenerate choices can give a constant or an identically zero equation instead of a straight line.

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