Petar's equations

Algebra Level 5

Consider the system of equations:

\[ \begin{align} xa + y\sin \left(\pi \sqrt{a^2+b^2}\right) & = 0 \\ xb + y\sin \left(\pi \sqrt{a^2+b^2}\right) &= 0 . \end{align}\]

How many different ordered pairs of integers \((a, b)\) subject to \(\left| a \right| < 6, \left| b \right| < 6\) are there such that the equations have a non-trivial solution \((x,y) \)?

This problem is posed by Petar V.

Details and assumptions

A non-trivial solution is a pair of values \( (x,y) \neq (0,0) \) which satisfies both equations.

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