 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2022/paper-1/11/c/iii/solution

i

is a counterclockwise rotation through

π /2

: it maps coordinates

(X, Y)

(- Y, X)

. The centre

(2, 2)

therefore becomes

(- 2, 2)

, and the long axis becomes vertical. For

w = X + iY

, the resulting ellipse is

\frac{( X + 2 ) ^{2}}{8 ^{2}} + \frac{( Y - 2 ) ^{2}}{1 6 ^{2}} = 1.

Its horizontal semiaxis is

8

and its vertical semiaxis is

16

, as shown in the final panel of the figure in part (b).

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