Among all the tangents of the curve X = t, y = - T ^ 2, z = T ^ 3, how many are parallel to the plane x + 2Y + Z = 4? Hello teacher, my solution is like this: 1 find the tangent vector (1, - 2T, 3T ^ 2) 2 orthogonal to the normal vector of the plane (1,2,1) can get 1-4t + 3T ^ 2 = 0, the solution is t = 1 / 3, t = 1, so there are two, but what I don't understand is why the answer adds a restriction, that is, after finding t, bring in the curve to get the tangent point P1 (1 / 3, - 1 / 9, 1 / 27), P2 (1, - 1, 1) Because the tangent point is not on the plane, the final answer is two. What I want to ask is that even if the tangent point is on the plane, the tangent vector within the plane is parallel to the plane. Why should it be regarded as a limiting condition

Among all the tangents of the curve X = t, y = - T ^ 2, z = T ^ 3, how many are parallel to the plane x + 2Y + Z = 4? Hello teacher, my solution is like this: 1 find the tangent vector (1, - 2T, 3T ^ 2) 2 orthogonal to the normal vector of the plane (1,2,1) can get 1-4t + 3T ^ 2 = 0, the solution is t = 1 / 3, t = 1, so there are two, but what I don't understand is why the answer adds a restriction, that is, after finding t, bring in the curve to get the tangent point P1 (1 / 3, - 1 / 9, 1 / 27), P2 (1, - 1, 1) Because the tangent point is not on the plane, the final answer is two. What I want to ask is that even if the tangent point is on the plane, the tangent vector within the plane is parallel to the plane. Why should it be regarded as a limiting condition

When there is no common point between two straight lines on the plane, two planes of space and one straight line and one plane of space, they are said to be parallel
The tangent vector in the plane does not belong to the stipulation that the line is parallel to the plane
Hope to adopt