Let S be the focus of the parabola y^2 = 12x . Two points P and Q lie on the parabola such that the centroid of the triangle PQS is G(21, 12) . The sum of the coordinates of the point of intersection of the tangents to the parabola at points P and Q is

Step-by-step solution

For the parabola y^2 = 12x , we have a = 3 . Thus, the focus is S(3, 0) . Let the parametric coordinates of P and Q be (3t₁^2, 6t₁) and (3t₂^2, 6t₂) respectively. The centroid of PQS is given as G(21, 12) . Using the centroid formula, for the x -coordinate: 3t₁^2 + 3t₂^2 + 3 3 = 21 t₁^2 + t₂^2 + 1 = 21 t₁^2 + t₂^2 = 20 For the y -coordinate: 6t₁ + 6t₂ + 0 3 = 12 2(t₁ + t₂) = 12 t₁ + t₂ = 6 We know the identity (t₁ + t₂)^2 = t₁^2 + t₂^2 + 2t₁t₂ . Substituting the values: (6)^2 = 20 + 2t₁t₂ 36 = 20 + 2t₁t₂ 2t₁t₂ = 16