Let p be a positive real, let S be the parabola y2 = 4px, and let P be a point with coordinates (a, b). Show that there are 1, 2 or 3 normals from P to S according as 4(2p - a)2 + 27 pb2 >, = or < 0.
The general point on the parabola is (pt2, 2pt). The slope of the tangent is 2p/y = 1/t, and so the slope of the normal is -t. Hence the equation of the normal is (y - 2pt) = -t(x - pt2). This passes through (a, b) iff pt3 + (2p - a)t - b = 0 (*).
This is a cubic, so it has 1, 2 or 3 real roots. We have to decide which. If (2p - a) > 0, then pt3 + (2p - a)t is strictly increasing and takes all values in (-∞, ∞), so (*) has just one real root. The same is true if (2p - a) = 0, unless b = 0, in which case there are three coincident roots.
If (2p - a) < 0, then pt3 + (2p - a)t has a maximum and a minimum. Differentiating, we find that these are at t = ± √( -(2p - a)/(3p) ), with values ± 2(2p - a)/3 √( -(2p - a)/(3p) ). So there are 1, 2 or 3 real roots according as b2 >, = or < 4(2p - a)2/9 -(2p - a)/3p, which is the condition in the question. Note that if (2p - a) ≥ 0, then this expression is certainly > 0, so the same rule applies.
3rd Putnam 1940
© John Scholes
15 Sep 1999