Putnam 1940 B3

Let $p>0$ be a real constant. From any point $(a,b)$ in the cartesian plane, show that i) Three normals, real or imaginary, can be drawn to the parabola $y^2=4px$ . ii) These are real and distinct if $4(2-p)^3 +27pb^2<0$ . iii) Two of them coincide if $(a,b)$ lies on the curve $27py^2=4(x-2p)^3$ . iv) All three coincide only if $a=2p$ and $b=0$ .