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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/ia/paper-1/1c/b/solution

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Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 1 / 1C / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-07

Put

z = x + i y

. Squaring the two nonnegative moduli and expanding gives

x^{2} + (y + 3)^{2} = 9 (x^{2} + y^{2})

. Completing the square yields

x^{2} + (y - \frac{3}{8})^{2} = (\frac{9}{8})^{2} .

(1)

Thus the locus is an Apollonius circle with center $3 i /8$ and radius $9/8$ . No extraneous points were introduced by squaring, because both original sides are nonnegative. Its intersections with the imaginary axis are

- 3 i /4

and

3 i /2

, useful landmarks for the sketch. It surrounds zero but does not pass through zero:

Figure 1.
Apollonius circle centered at three eighths i with radius nine eighths and imaginary-axis intercepts minus three quarters i and three halves i
.

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