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

Write

z = x + i y

z^{*} = x - i y

. Then

u = \frac{3}{2} (x + i y) + \frac{1}{2} (x - i y) = 2 x + i y .

Thus the real part is stretched by a factor of two while the imaginary part is unchanged. If

u = X + iY

, substitution into the circle equation gives

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

This is an ellipse with centre

(- 2, - 1)

, horizontal semiaxis

16

and vertical semiaxis

8

. It is the second panel in the figure in part (b).

ii

Adding

4 + 3 i

(4, 3)

in the complex plane. The ellipse of part (i) therefore keeps its semiaxes and axis directions, with new centre

(2, 2)

v = X + iY

, its equation is

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

The third panel in the figure in part (b) shows the translated ellipse.

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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