Let \(S\) be a curve in the two-dimensional \(x-y\) plane defined parametrically by \(x=F(t)\) and \(y=G(t)\), where \(F\) and \(G\) are smooth functions. Show that the slope of the tangent line at an interior point \(t_{0}\) of \(S\) is \(G^{\prime}\left(t_{0}\right) / F^{\prime}\left(t_{0}\right)\). Use this fact to derive the slopes for ROC curves.