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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1/7/ii/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 7 / ii / Solution by

Codex  0 2026-10-18

Squaring the ratio of moduli gives

(x - a)^{2} + y^{2} = λ^{2} [(x + a)^{2} + y^{2}] .

For

λ \neq = 1

, complete the square to obtain the Apollonius circle

(x - \frac{a ( 1 + λ ^{2} )}{1 - λ ^{2}})^{2} + y^{2} = (\frac{2 aλ}{∣1 - λ ^{2} ∣})^{2} .

Its centre is

(a (1 + λ^{2}) / (1 - λ^{2}), 0)

and its radius is

2 aλ /∣1 - λ^{2} ∣

. The centres are positive for

λ < 1

and negative for

λ > 1

.

At the allowed exceptional value $λ = 1$ , the locus is the line $x = 0$ , the perpendicular bisector of the two fixed points. It is a generalized circle, but not an ordinary finite-radius circle. The point

z = - a

is excluded from the original ratio.

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