Vector Functions and Space Curves PLEASE HELP!

Question

At what points does the curve r(t) = ti + (4t − t^2)k intersect the paraboloid z = x^2 + y^2?
(If an answer does not exist, enter DNE.)

 

smaller t-value (x, y, z) =

larger t-value (x, y, z) = 

 

Please explain and thanks in advance!!

Answer 1

The space curve is

x=t

y=0

z=4t-t².

 

When does this satisfy the equation z = x²+y²?

 

4t-t² = t²+0

2t(t-2)=0

2t = 0  or  t-2 = 0

 

t = 0  or t = 2

Which give points (0,0,0)  and  (2,0,4)

 

I hope that helps