Let P be a point on the parabola y^2 = 8x and O be its vertex. The normal to the parabola at the point P meets the x -axis at the point N . The equation of the locus of the centroid of the triangle OPN is :

Step-by-step solution

The equation of the parabola is y^2 = 8x , so a = 2 . Let the parametric coordinates of the point P be (2t^2, 4t) . The equation of the normal to the parabola at P is given by: y = -tx + 2at + at^3 y = -tx + 4t + 2t^3 Since the normal meets the x -axis at N , we set y = 0 : 0 = -tx + 4t + 2t^3 Assuming t 0 , dividing by t gives: x = 4 + 2t^2 Thus, the coordinates of N are (2t^2 + 4, 0) . Let G(h, k) be the centroid of the triangle OPN . The coordinates of the vertices are O(0, 0) , P(2t^2, 4t) , and N(2t^2 + 4, 0)