Two differentiable curves are given by the relations F(x,y)=0 and G(x,y)=0.
(i) State what it means for the two curves to have a common tangent at a point P(x0,y0) lying on both curves.
(ii) Show that, if P(x0,y0) satisfies both relations, equality of the slopes (dxdy) calculated separately for each curve at P is a necessary condition for the curves to possess a common tangent at P.
(iii) Prove that this condition is also sufficient provided the two slopes are finite and the two curves are differentiable at P.