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 first quadrant (and hence, depend on one parameter only). Find values for which piece corresponding parabola either encloses minimum area, or has length. We find sufficient condition under fixed point, area minimizing curve and length coincide. The problem led us to certain implicit function we explored its asymptotic behavior convexity.