Curves & Motion · Challenge problem

A ladder of length $5$ slides with its top on the $y$-axis and its foot on the positive $x$-axis. A paint mark on the ladder, $2$ units from the top, traces a curve as the ladder slides. Find the radius of curvature of that curve at the moment the ladder makes a $45^\circ$ angle with the ground. Give an exact answer in simplest radical form.