In this paper, the necessary and sufficient conditions for minima plane path with a movable end-point are developed. Using the calculus of variations the considered conditions are based on the zero first-order nonsimultaneous variation and on the positive second-order variation in the functional of integral type corresponding to mechanical systems. The applied procedure is the coordinate parametric method. The obtained solutions are tested on a brachistochrone with one end-point constrained to lie on a circle. The exact solution is compared with the approximate one obtained with Ritz’s method.