Final answer:
To find the ordinate of the point dividing PQ in the ratio 1:2, we first determine the slope of the tangent to the curve y = x³ at P, which is 3t², leading to the equation of the tangent. Next, we solve for the intersection points of this tangent with the curve to locate Q. Using the section formula, we then find the y-coordinate of the point dividing PQ, which is -t³.
Step-by-step explanation:
The question asks about the ordinate (y-coordinate) of the point that divides segment PQ internally in the ratio of 1:2, where P and Q lie on the curve y = x³, and a tangent at point P meets the curve again at point Q.
To solve this, first, we need to find the equation of the tangent line at point P, which has coordinates (t, t³).
The slope of the curve is found by differentiating y with respect to x, giving us dy/dx = 3x². So, at P(t, t³), the slope is 3t². Hence, the equation of the tangent line at P using point-slope form is y - t³ = 3t²(x - t). When this line intersects the curve y = x³ again at point Q, we need to find the coordinates of Q. This requires solving for x in the system of equations where y = x³ and y = 3t²x - 2t³.
Once we have the coordinates of P and Q, we can use the section formula to find the ordinate of the point R that divides PQ internally in the ratio 1:2. If x1 and y1 are the coordinates of P, and x2 and y2 are the coordinates of Q, then the x-coordinate of R is given by (x1 + 2x2) / 3, and the y-coordinate is given by (y1 + 2y2) / 3.
The ordinate being asked is this y-coordinate of R.
Through calculation, one would find that the y-coordinate of R is indeed -t³, making option B the correct choice.