Prove that the curves \(y^2 = 16x\) and \(y = x^2 + 3\) touch at the point \((1,4)\). Prove also that the common tangent at this point forms with the axes a triangle of unit area.

Prove that the curves \(y^2 = 16x\) and \(y = x^2 + 3\) touch at the point \((1,4)\). Prove also that the common tangent at this point forms with the axes a triangle of unit area.