Let x = 2 t , y = t 2 3 be a conic. Let S be the focus and B be the point on the axis of the conic such that S A ⊥ B A , where A is any point on the conic. If k is the ordinate of the centroid of the Δ S A B , then lim t → 1 k is equal to

Step-by-step solution

The parametric form x = 2 t , y = t 2 3 represents the parabola x 2 = 12 y Given A S ⊥ A B So, m A S · m A B = - 1 ⇒ 3 - t 2 3 0 - 2 t · α - t 2 3 0 - 2 t = - 1 ⇒ 3 α = 27 t 2 + t 4 t 2 - 9 Ordinate of centroid of Δ S A B = k = α + t 2 3 + 3 3 = 9 + 3 α + t 2 9 lim t → 1 k = lim t → 1 1 9 9 + t 2 + 27 t 2 + t 4 t 2 - 9 = 13 18