Say $a^2+b^2=c^2$ is a primitive Pythagorean triple. Then consider the Linear Diophantine Equation $$ua^2+vb^2+xab+ybc+zca=0$$ where $(u,v,x, y, z)\in\mathbb Z^4$ are variables. If $(u,v,x, y, z)\neq(0,0,0,0,0)$ then can we say anything about $\|(u,v,x, y, z)\|_\infty$ or the probability distribution of $\|(u,v,x, y, z)\|_\infty$?

By this I mean can $\|(u,v,x, y, z)\|_\infty$ be much smaller than $\sqrt{\max(a^2,b^2)}$?