Let A be the focus of the parabola y²=8 x . Let the line y= m x+ c intersect the parabola at two distinct points B and C . If the centroid of the triangle A B C is ( 7 3 , 4 3 ) , then (B C)² is equal to :

Step-by-step solution

Coordinates of centroid of triangle ABC are 2 3 (t₁^2 + t₂^2 + 1) = 7 3 t₁^2 + t₂^2 = 5 2 4 3 (t₁ + t₂) = 4 3 t₁ + t₂ = 1 (t₁ + t₂)^2 = t₁^2 + t₂^2 + 2t₁ t₂ t₁ t₂ = -3 4 (t₁ - t₂)^2 = (t₁ + t₂)^2 - 4t₁ t₂ = 4 (BC)^2 = 4(t₁^2 - t₂^2)^2 + 16(t₁ - t₂)^2 (BC)^2 = 80