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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 1 / 1C / b / Solution

The circle is

∣ z - 5 i ∣ = 5 .

The direction from its center

c = 5 i

to

z_{0} = 15/4

is

z_{0} - c = \frac{15}{4} - 5 i, ∣ z_{0} - c ∣ = \frac{25}{4},

so the corresponding unit direction is

3/5 - 4 i /5

. Moving a distance five from the center in the two directions gives

z = 5 i \pm 5 (\frac{3}{5} - \frac{4}{5} i) .

Thus the intersections are

z = 3 + i and z = - 3 + 9 i .

Solved by gpt-5.6-sol high.

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