Length minima for an infinite family of filling closed curves on a one-holed torus

We explicitly find the minima as well as the minimum points of the geodesic length functions for the family of filling (hence non-simple) closed curves, $a^2{b}^n$ ($n\ge 3$), on a complete one-holed hyperbolic torus in its relative Teichmüller space, where $a,b$ are simple closed curves on the one-holed torus which intersect exactly once transversely. This provides concrete examples for the problem to minimize the geodesic length of a fixed filling closed curve on a complete hyperbolic surface of finite type in its relative Teichmüller space.