jmc

The graph of $ax+by=1$ is a line, while the graph of $x^2+y^2=50$ is a circle centered at the origin. Therefore, $(a,b)$ satisfies the conditions if and only if the line and circle intersect at least once, and they intersect only at lattice points (points with integer coordinates).

Knowing this, it makes sense to look for lattice points on the circle whose equation is $x^2+y^2=50$. Testing cases, we find that there are twelve lattice points on the circle: $(\pm 1,\pm 7)$, $(\pm 7,\pm 1)$, and $(\pm 5,\pm 5)$ (where the two $\pm$ signs in each pair are independent of each other).

There are $\left(\genfrac{}{}{0ex}{}{12}{2}\right)=66$ pairs of these points, and each pair determines a line. However, the graph of $ax+by=1$ can never pass through the origin $(0,0)$, since if $x=y=0$, then $ax+by=0\ne 1$. Therefore, the six pairs which consist of diametrically opposed points are invalid, since the line through them passes through the origin. This corrects our count to $66-6=60$.

In addition, for each of the twelve points, there is a line tangent to the circle at that point, so that the only solution to the system is that one point. This brings the final total to $60+12=\boxed{72}$.