Let P be a point on the parabola y^2 = 8x with vertex O . A line L passing through O is drawn perpendicular to the tangent at P . If L intersects the parabola again at point Q , then the locus of the centroid of triangle OPQ is :

Step-by-step solution

For the parabola y^2 = 8x , a = 2 . Let the coordinates of P be (2t^2, 4t) . The slope of the tangent to the parabola at P is 2a y₁ = 4 4t = 1 t . Since line L is perpendicular to the tangent at P , its slope is -t . Line L passes through the origin O(0,0) , so its equation is: y = -tx To find the coordinates of Q , substitute y = -tx into the equation of the parabola: (-tx)^2 = 8x t^2 x^2 = 8x Since Q is not the origin, x 0 , giving x = 8 t^2 . Then, y = -t ( 8 t^2 ) = - 8 t . Thus, Q = ( 8 t^2 , - 8 t ) . Let G(h