Intersection of a Curve and a Surface

[major_maths]

1. At what points does the curve r(t)=ti+2tj+t²k intersect the surface z = x²+y²-100? Give the coordinates of the points.

2. Given equations above.

3. r(t)=<t, 2t, t²>
z = x²+y²-100
(t²) = (t)²+(2t)-100
-4t² = -100
t = sqrt(25) = +/- 5

when t = 5, (5, 10, 25)
when t = -5 (-5, -10, 25)

This was a problem on a test, and when I got it back my teacher gave me zero credit. So, I don't think I know how to do this sort of problem but I'm coming up with a blank when I try to solve it a different way.

 

[Mark44]

1. At what points does the curve r(t)=ti+2tj+t²k intersect the surface z = x²+y²-100? Give the coordinates of the points.

2. Given equations above.

3. r(t)=<t, 2t, t²>
z = x²+y²-100
(t²) = (t)²+(2t)-100

Should be
t² = (t)²+(2t)²-100

-4t² = -100
t = sqrt(25) = +/- 5

sqrt(25) means the principal square root of 25, which is 5.
Your equation above should be t = +/-sqrt(25) = +/- 5

when t = 5, (5, 10, 25)
when t = -5 (-5, -10, 25)

I get the same points. Maybe your instructor marked off because of the errors I pointed out. It might be worthwhile to ask your instructor why you got no credit for the problem.

r(5) = <5, 10, 25> and (5, 10, 25) is a point on the surface (which happens to be a paraboloid).
r(-5) = <-5, -10, 25) and (-5, -10, 25) is another point on the surface.

This was a problem on a test, and when I got it back my teacher gave me zero credit. So, I don't think I know how to do this sort of problem but I'm coming up with a blank when I try to solve it a different way.