If the chord joining the points has parameters t1 and t2 pass through the focus, then t1t2 = − 1. In other words, if the coordinates of one end of a focal chord are (at2, 2at) then the coordinates of the other end are (a/t2, -2a/t).
The tangent at the extremities of any focal chord of a parabola intersects at a right angle on the directrix.
The locus of the point of intersection of the tangents perpendicular to each other of the parabola y2 = 4ax is its directrix. This is also called as director circle of the parabola.
The area of the triangle formed by three points on a parabola is twice the area of the triangle formed by the tangents at these points.
If the normal y + t1x = 2at1 + at13 at the point (at12, 2at1) meets the parabola y2 = 4ax again at t2, then t2 = −t1 − 2/t1.
If the normals at points t1 and t2 meet at a point t3 on the parabola y2 = 4ax then t1t2 = 2, and t3 = −(t1 + t2).
Semilatus rectum of the parabola y2 = 4ax is the harmonic mean between the segments of any focal chord of the parabola.
The centroid of the triangle formed by the feet of the normal points lies on the axis of the parabola.
The chord joining the points of contact of a pair of perpendicular tangents passes through the focus of the parabola.
The tangent and normal at any point P of a parabola meet the axis in T and Q then
ST = SQ = SP.
The tangent at P is equally inclined to the axis and the focal distance.
∠ PSK is a right angle where K is the point where the tangent at P meets the directrix.
