parametric equation problem. i need help!!?

Question

Given the curve with parametric equations x= t^2 - 1, y= 1-t^3, z=1-2t+t^2

Does it pass through the point A(3,-7,1)?

Answer 1

Sub (3,-7,1) in those parametric equations, see whether they give the same 't'. If they do, the answer is 'yes'.

Answer 2

3 = t^2 - 1
4 = t^2
t = +2 or - 2

Now plug in +2 or -2 in the y equation and see if you get -7. Let's try

y = 1 - t^3
-7 = 1 - (2^3) = -7 checks.
-7 = 1 - (-2^3) = 9 does not check.

Ok, so we got -7 with t = 2.

If t = 2 for z gives us 1 then it passes through A(3,-7,1) @ t = 2

1 = 1 - 2t + t^2
1 = 1 - 4 + 4 = 1

I guess it does pass through (3, -7, 1)

Answer 3

Plug in the point and solve for t. If all the variables give the same value for t, the curve passes thru A(3, -7, 1).

x = 3 = t² - 1
t² = 4
t = ±2

y = -7 = 1 - t³
t³ = 8
t = 2

z = 1 = 1 - 2t + t²
t² - 2t = 0
t(t - 2) = 0
t = 0, 2

The value t = 2 works for each of the variables so the curve does pass thru A(3, -7, 1).

Answer 4

dy/dx=(dy/dt)/(dx/dt)=(-2/t^2)/-2=a million/t^2 For element A, a million-2t=3 so t=-a million and dy/dx at A = a million/(-a million)^2=a million. Slope of standard = -a million/(dy/dx)=-a million so equation of standard is (y--4)=-a million(x-3) or y+x+a million=0. Sub parametric form of curve into y+x+a million=0, giving -2+2/t + a million-2t+a million=0 and those cleans as much as two-2t^2=0 or t^2=a million giving t=a million and t=-a million. t=-a million is A so t=a million is B with coordinates x=a million-2=-a million and y=-2+2/a million=0

Answer 5

Northstar is correct.