Question

Prove that the tangents at the extremities of a focal chord of a parabola intersect at right angles

Answer 1

Let the parabola be y2 = 4ax Equation of the tangent at P(t1) is Equation of the tangent at Q(t2) is Solving, point of intersection is Equation of the chord PQ is Since PQ is a focal chord, S (a,0) is a point on PQ. Therefore, 0 = 2a +2a t1 t2 ⇒ t1 t2 = -1. Therefore point of intersection of the tangents is [ a,a(t1+ t2)] The x coordinate of this point is a constant. And that is x = -a which is the equation of the directrix of the parabola.