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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/2004/paper-1/10/b/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 / 2004 / Paper 1 / 10 / b / Solution by

Codex  0 2026-10-20

The modulus equation says that

z

has fixed distance eight from

- 1 - i

. Thus its locus is a circle of radius eight centered at $(- 1, - 1)$ , with Cartesian equation

(x + 1)^{2} + (y + 1)^{2} = 64

.

Figure 1.
The original circle, a horizontal stretch, a translation and a quarter-turn
. Original complex-plane loci. The map $u = 2 x + i y$ stretches only the horizontal coordinate; $v = z + 4 + 3 i$ translates the circle; $w = i v$ rotates that translated circle about the origin. Marked dots show each center. Details.

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