= Solution
Put $z=x+iy$. Squaring the two nonnegative moduli and expanding gives $x^2+(y+3)^2=9(x^2+y^2)$. Completing the square yields
$$
\boxed{x^2+\left(y-\frac38\right)^2=\left(\frac98\right)^2.}
$$
Thus the locus is an <Apollonius circle> with \b[center $3i/8$ and radius $9/8$]. No extraneous points were introduced by squaring, because both original sides are nonnegative. Its intersections with the imaginary axis are $-3i/4$ and $3i/2$, useful landmarks for the sketch. It surrounds zero but does not pass through zero:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/ia/paper-1-apollonius-circle.png]
{title=Apollonius circle centered at three eighths i with radius nine eighths and imaginary-axis intercepts minus three quarters i and three halves i}
Back to article page
