Parametric curve question (determining unknown point)

  • #1
cherry
17
6
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:
IMG_5937C097F81C-1.jpeg


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! :)
 
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  • #2
cherry said:
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?
 
  • #3
cherry said:
I see where I went wrong and it turns out t = -1 and k = 2 is the correct solution.
Hello @cherry, and
:welcome: ##\qquad## !​

Kudos for finding out!
1705184825929.png
is indeed 12, not 16. (*)

cherry said:
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.

##\ ##
 

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