= Solution
The square roots simplify to the <absolute values> $|a-b|$ and $|a+b|$. Put $s=\sin(\pi\sqrt{a^2+b^2})$. A nontrivial solution of the homogeneous <linear system> exists exactly when its <coefficient matrix> is singular:
$$
0=\det\begin{pmatrix}|a-b|&s\\|a+b|&s\end{pmatrix}
=s(|a-b|-|a+b|).
$$
The second factor vanishes precisely when $(a-b)^2=(a+b)^2$, or $ab=0$, giving both coordinate axes. The first factor vanishes when $\sqrt{a^2+b^2}$ is a nonnegative integer. Hence the complete <locus> is
$$
\boxed{\{a=0\}\ \cup\ \{b=0\}\ \cup\
\bigcup_{n=1}^{\infty}\{a^2+b^2=n^2\}.}
$$
These are the two axes together with concentric <circles> of positive integer radii. The zero-radius case is the origin, already included on the axes; at that point both equations vanish and any nonzero pair $(x,y)$ works.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-1-singular-system-locus.png]
{title=The parameter locus consists of both axes and circles with integer radii}
Back to article page
