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 Incoming links: Affine image of a circle

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 1 / 10 / b / i / Solution 2026-10-20

u = X + iY

gives

X = 2 x

,

Y = y

. Substitute

x = X /2

,

y = Y

into the original circle equation. This affine image of a circle is the ellipse

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

centered at

(- 2, - 1)

with horizontal semiaxis sixteen and vertical semiaxis eight. The map covers the whole ellipse as

z

covers the circle.

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