Normal Lines Drawn to a Parabola and Geometric Constructions

Abstract

Using the discriminant of a certain cubic polynomial, we will obtain a necessary and sufficient condition on the coordinates of the point P0 in order that the number of normal lines from P0 to the parabola is either one, two or three. This will lead to the observation that there are only two points A and B on the parabola with the property that exactly two normal lines can be drawn from either A or B to the parabola. Except A and B, from any other point on the parabola, one can draw either exactly one or three normal lines. Therefore, in a certain sense, these points A and B serve as special cut-off points on the parabola. The following theorem will partly reveal their special nature:

Theorem 1.
Consider an arbitrary parabola and all the circles passing through its vertex V. Then only two such circles touch the parabola at a point P1 different from V. One of these circles pass through the forementioned point A on the parabola, and the other circle passes through the point B on the parabola.

Suppose for a moment that the point P0(x0,y0) is on the parabola y^2=4ax with x0>8a. Let Q and R be the feet of the normals drawn from P0 to the parabola. We have made several computations, to find the area, centroid G, orthocenter H, and the circumcenter C of the triangle P0QR. As the point P0 moves on the parabola, the loci of the points G, H and C are very interesting. We have made several Mathematica animations to demonstrate these loci.

We will obtain several more theorems:

Theorem 2 Let Pi (ati^2, 2ati) i=1,2,3 be any three distinct points on the parabola y^2=4ax. Then the following are equivalent: 1. t1+t2+t3=0. 2. The normals to the parabola at the points Pi are concurrent. 3. The circle through the points Pi (i=1,2,3) pass through the vertex of the parabola.

Theorem 3 Let Pi (ati^2, 2ati) i=1,2,3 be any three distinct points on the parabola y^2=4ax such that the normals at Pi are concurrent. Then the point of concurrency D is given by (a ( 2 + t1^2 + t1*t2 + t3^2, -ati*t2 (t1 + t2) ). Further, the orthocenter of the triangle P1P2P3 and the circumcenter of the triangle formed by the tangents at Pi (i=1,2,3) are equidistant from the axis of the parabola. Moreover, the orthocenter of of the triangle formed by the tangents at Pi (i=1,2,3) and the point of concurrency D are equidistant from the axis.

Theorem 4 Consider a circle centered at (u,v) passing through the vertex V of the parabola y^2=4ax, and intersecting the parabola at three more distinct points Pi (i=1,2,3) other than V. Then the normals at the points Pi are concurrent at ( 2 (u - a), 4v ).

As a byproduct of the paper, the last theorem above provides a beautiful geometric construction of all the normal lines from any point on the plane to an arbitrary parabola.

We will also consider several probability issues: For example, we will compute the probability of having exactly three normal lines from a point on the plane to a given parabola, if that point is randomly chosen from a disk centered at the origin with a given radius.