If the tangent to the curve y = x 3 at the point P t , t 3 meets the curve again at Q , then the ordinate of the point which divides P Q internally in the ratio 1 : 2 is:

Step-by-step solution

Slope of tangent at P t , t 3 = d y d x t , t 3 = 3 x 2 x = t = 3 t 2 So equation tangent at P t , t 3 : y - t 3 = 3 t 2 x - t for point of intersection with y = x 3 x 3 - t 3 = 3 t 2 x - 3 t 3 ⇒   x - t x 2 + x t + t 2 = 3 t 2 x - t for x ≠ t x 2 + x t + t 2 = 3 t 2 ⇒   x 2 + x t - 2 t 2 = 0 ⇒ x - t x + 2 t = 0 So for Q : x = - 2 t , Q - 2 t , - 8 t 3 ordinate of required point : 2 t 3 - 8 t 3 2 + 1 = - 2 t 3