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

Writing $u=X+iY$ gives $X=2x$, $Y=y$. Substitute $x=X/2$, $y=Y$ into the original circle equation. This affine image of a circle is the ellipsecentered at $(-2,-1)$ with horizontal semiaxis sixteen and vertical semiaxis eight. The map covers the whole ellipse as $z$ covers the circle.