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 y2=4px.
ii) These are real and distinct if 4(2−p)3+27pb2<0.
iii) Two of them coincide if (a,b) lies on the curve 27py2=4(x−2p)3.
iv) All three coincide only if a=2p and b=0. Putnamanalytic geometryconicsparabola