Existence and Uniqueness of a Curve with Both Minimal Length and Minimal Area

Consider the family of generalized parabolas {y=−axr+c|a,r,c,x>0,risafixedconstant} that pass through a given point in the first quadrant (and hence, depend on one parameter only). Find the parameter values for which the piece of the corresponding parabola in the first quadrant either encloses a minimum area, or has a minimum length. We find a sufficient condition under which given the fixed point, the area minimizing curve and the length minimizing curve coincide. The problem led us to a certain implicit function and we explored its asymptotic behavior and convexity