 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/7/a/i/solution

Write

z = x + i y

and square the modulus equation. It becomes

x^{2} + (y + b)^{2} = P^{2} [x^{2} + (y - b)^{2}] .

For

P \neq = 1

x^{2} + (y - \frac{b ( P ^{2} + 1 )}{P ^{2} - 1})^{2} = \frac{4 b ^{2} P ^{2}}{( P ^{2} - 1 ) ^{2}} .

Hence

center (0, \frac{b ( P ^{2} + 1 )}{P ^{2} - 1}), radius \frac{2 b P}{∣ P ^{2} - 1∣} .

The printed statement needs the exception

P = 1 ⟹ y = 0

: equal distances from

ib

and

- ib

P > 1

the center lies in the upper half-plane; this is the case needed in part (ii). The original PDF confirms the numerator

z + ib

, correcting the duplicated, corrupted OCR formula.

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