Tangents are drawn at the endpoints of a focal chord of the parabola y^2 = 12x , which intersect at a point R . If the length of the focal chord is 27 , then the square of the perpendicular distance from the point R to the focal chord is

Step-by-step solution

For the parabola y^2 = 12x , we have a = 3 . Let the endpoints of the focal chord be P(at^2, 2at) and Q ( a t^2 , - 2a t ) . The tangents at the extremities of a focal chord intersect on the directrix at the point R (-a, a (t - 1 t ) ) . The equation of the focal chord PQ passing through (a,0) with slope 2 t - 1 t is: 2x - (t - 1 t )y - 2a = 0 The perpendicular distance d from R to the focal chord PQ is: d = |2(-a) - (t - 1 t )a (t - 1 t ) - 2a | 4 + (t - 1 t )^2 d = |-4a - a (t - 1 t )^2 | (t + 1 t )^2 = a (4 + (t