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$r(t) = ti+(2t-t^{2})k$ intersect the paraboloid $z = x^2 + y^2$

What am I missing here? Can I get some hints that lead me as to what I need to do here? I haven't the faintest idea where to start. I thought maybe write the vector equation as a parametric equation then solve for t and try to set the equations equal but that does not lead me to any points. As I still have two variable y and x.

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1 Answer

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Hint:

the points of the curve have coordinates: $$ [x,y,z]=[t,0,2t-t^2] $$

so, the common points with the paraboloid are such that: $$ z=x^2+y^2 \quad \iff \quad2t-t^2=t^2 $$

find $t$ from this equation and you have the points.

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