Putnam 1965 A6

A-6. In the plane with orthogonal Cartesian coordinates $x$ and $y$, prove that the line whose equation is $u x+v y=1$ will be tangent to the curve $x^{m}+y^{m}=1$ (where $m>1$ ) if and only if $u^{n}+v^{n}=1$ and $m^{-1}+n^{-1}=1$.