 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/c/iii/solution

For the translation, expand the products of complex numbers and collect the terms in

w \overset{w}{ˉ}

\overset{w}{ˉ}

w

and the constant:

a (w + α) (\overset{w}{ˉ} + \overset{α}{ˉ}) + b (\overset{w}{ˉ} + \overset{α}{ˉ}) + \overset{ˉ}{b} (w + α) + c = 0.

A = a, B = a α + b, C = a ∣ α ∣^{2} + b \overset{α}{ˉ} + \overset{ˉ}{b} α + c .

The last coefficient is real because

b \overset{α}{ˉ} + \overset{ˉ}{b} α = 2 Re (\overset{ˉ}{b} α)

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