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

The modulus equation says that $z$ has fixed distance eight from $-1-i$. Thus its locus is a circle of radius eight centered at $(-1,-1)$, with Cartesian equation $(x+1{)}^2+(y+1{)}^2=64$.

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.

The complex translation adds four to the real coordinate and three to the imaginary coordinate. The locus is therefore a circle centered at $(3,2)$ with radius eight:

Multiplication by $i$ rotates the plane counterclockwise through $\pi /2$. The center $3+2i$ maps to $-2+3i$, while distances are preserved. Thus the locus is a circle centered at $(-2,3)$ with radius eight: