 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/11/b/iii/solution

A squared complex modulus is nonnegative. Conversely, every nonnegative right-hand side of

∣ z + b ∣^{2} = ∣ b ∣^{2} - c

can be attained. The necessary and sufficient condition is therefore

c \leq ∣ b ∣^{2} = b_{1}^{2} + b_{2}^{2} .

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