Parabola - Orthoptic Property

Orthoptic Property

If two tangents to a parabola are perpendicular to each other, then they intersect on the directrix. Conversely, two tangents which intersect on the directrix are perpendicular.

Proof

Without loss of generality, consider the parabola Suppose that two tangents contact this parabola at the points and Their slopes are and respectively. Thus the equation of the first tangent is of the form where is a constant. In order to make the line pass through the value of must be so the equation of this tangent is Likewise, the equation of the other tangent is At the intersection point of the two tangents, Thus Factoring the difference of squares, cancelling, and dividing by 2 gives Substituting this into one of the equations of the tangents gives an expression for the y-coordinate of the intersection point: Simplifying this gives

We now use the fact that these tangents are perpendicular. The product of the slopes of perpendicular lines is -1, assuming that both of the slopes are finite. The slopes of our tangents are and, so so Thus the y-coordinate of the intersection point of the tangents is given by This is also the equation of the directrix of this parabola, so the two perpendicular tangents intersect on the directrix.