Let C: F(X,Y,Z) = 0 be a projective curve given by a homogeneous poly- nomial F C[X, Y, Z], and let P = P2 be a point. (a) Prove that P is a singular point of C if and only if OF x (P) = 3F (P) = F ax OF az (P) = 0. (b) If P is a non-singular point of C, prove that by the equation OF (P)X + ay (P)Y+ the tangent line to C at P is given OF az (P)Z = 0.