 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/11/d/ii/solution

The same completed square gives

∣ z + b ∣^{2} = ∣ b ∣^{2} - c

. A nondegenerate circle therefore occurs exactly when

c < ∣ b ∣^{2}, centre - b, radius ∣ b ∣^{2} - c .

c > ∣ b ∣^{2}

gives an empty locus because a squared complex modulus cannot be negative.

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