Let S be the focus of the parabola y^2 = 8x . Let A be a point on the parabola ( A (0,0) ) and B be a point on the x-axis such that SA AB . If the abscissa of the centroid of SAB is 86 9 , then the sum of all possible values of the abscissa of A is

Step-by-step solution

For the parabola y^2 = 8x , 4a = 8 a = 2 . The focus is S(2, 0) . Let the coordinates of A be (2t^2, 4t) and B be (x_B, 0) . The slope of SA is m_ SA = 4t - 0 2t^2 - 2 = 2t t^2 - 1 . The slope of AB is m_ AB = 0 - 4t x_B - 2t^2 = -4t x_B - 2t^2 . Since SA AB , m_ SA m_ AB = -1 : ( 2t t^2 - 1 ) ( -4t x_B - 2t^2 ) = -1 8t^2 (t^2 - 1)(x_B - 2t^2) = 1 x_B - 2t^2 = 8t^2 t^2 - 1 x_B = 2t^2 + 8t^2 t^2 - 1 = 2t^4 - 2t^2 + 8t^2 t^2 - 1 = 2t^4 + 6t^2 t^2 - 1 . The abscissa of the centroid of SAB is given by x_S + x_A + x_B 3