Problem E

The Shortest Path

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 \(10\). Letâ€™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 \(10\).