On the Cartesian plane, we draw grid lines at integer points along the \(x\) and \(y\) axes. We can then draw paths along these grid lines between any two points with integer coordinates. The graph below shows two paths along these grid lines from \(O(0,0)\) to \(P(6,-4)\). One path has length \(10\) and the other has length \(20\).
There are many different paths along the grid lines from \(O\) to \(P\), but the smallest possible length of such a path is \(10\). Let’s call this smallest possible length the path distance from \(O\) to \(P\).
Determine the number of points with integer coordinates for which the path distance from \(O\) to that point is \(10\).