Let PQ be a focal chord of the parabola y^2 = 16x of length 25 , and let V be the vertex of the parabola. If G( , ) is the centroid of the triangle PVQ with > 0 , then the value of 3 + ^2 is

Step-by-step solution

For the parabola y^2 = 16x , we have a = 4 . The vertex is V(0, 0) . Let the endpoints of the focal chord be P(4t^2, 8t) and Q ( 4 t^2 , - 8 t ) . The length of a focal chord is given by a (t + 1 t )^2 . Given that the length is 25 : 4 (t + 1 t )^2 = 25 (t + 1 t )^2 = 25 4 We can find t^2 + 1 t^2 as follows: t^2 + 1 t^2 = (t + 1 t )^2 - 2 = 25 4 - 2 = 17 4 We also need t - 1 t to find the y -coordinate of the centroid: (t - 1 t )^2 = (t + 1 t )^2 - 4 = 25 4 - 4 = 9 4 t - 1 t = 3 2 The centroid G( , ) of PVQ is: = 4