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Putnam 1940 B3

Source: Putnam 1940

February 22, 2022
Putnamanalytic geometryconicsparabola

Problem Statement

Let p>0p>0 be a real constant. From any point (a,b)(a,b) in the cartesian plane, show that i) Three normals, real or imaginary, can be drawn to the parabola y2=4pxy^2=4px. ii) These are real and distinct if 4(2−p)3+27pb2<04(2-p)^3 +27pb^2<0. iii) Two of them coincide if (a,b)(a,b) lies on the curve 27py2=4(x−2p)327py^2=4(x-2p)^3. iv) All three coincide only if a=2pa=2p and b=0b=0.
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