 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-1/11/c/solution

With

z = x + i y

, the expression inside the modulus is

x + 3 i (y - 1)

. Squaring its nonnegative modulus therefore gives

x^{2} + 9 (y - 1)^{2} = 4, or \frac{x ^{2}}{4} + \frac{( y - 1 ) ^{2}}{4/9} = 1.

The locus is an ellipse in the complex plane, centred at

(0, 1)

, with horizontal semiaxis

2

and vertical semiaxis

2/3

. Its vertices are

(\pm 2, 1)

and

(0, 1 \pm 2/3)

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