Parametric curve question (determining unknown point)

[cherry]

Homework Statement

A curve given parametrically by (x, y, z) = (3 - t, -1 - 3t^2, 2t + 2t^3). There is a unique point P on the curve with the property that the tangent line at P passes through the point (2, 8, 12). What are the coordinates of point P?

Relevant Equations

(x, y, z) = (3 - t, -1 - 3t^2, 2t + 2t^3)

My work so far:

I am stuck because when I inputted the two possible values of t and k, neither solution worked. Where did I go wrong? Pointers would be appreciated! :)

 

[cherry]

Homework Statement: A curve given parametrically by (x, y, z) = (3 - t, -1 - 3t^2, 2t + 2t^3). There is a unique point P on the curve with the property that the tangent line at P passes through the point (2, 8, 12). What are the coordinates of point P?
Relevant Equations: (x, y, z) = (3 - t, -1 - 3t^2, 2t + 2t^3)

My work so far:
View attachment 338514

I am stuck because when I inputted the two possible values of t and k, neither solution worked. Where did I go wrong? Pointers would be appreciated! :)

I see where I went wrong and it turns out t = -1 and k = 2 is the correct solution.
Where would I go from there to determine point P?

 

[BvU]

I see where I went wrong and it turns out t = -1 and k = 2 is the correct solution.

Hello @cherry, and

##\qquad## !​

Kudos for finding out!

is indeed 12, not 16. (*)

Where would I go from there to determine point P?

You have ##(x, y, z) = (3 - t\; , -1 - 3t^2\; , 2t + 2t^3) \ !##(*) quoting is a lot easier if ##\LaTeX## is used. See link to guide at lower left of edit window...

[edit] I didn't check if k=2 is the correct solution, nor whether the other solution is invalid
[edit] did now.

##\ ##