At what points does the curve $ r(t) = ti + (2t - t^2) k $ intersect the paraboloid $ z = x^2 + y^2 $?
so in this Probably want to determine. Uh, at what points or point Do the curve? Does the curve are of tea? Equal T I plus to t minus t squared times k intersect the problem oId z equals X squared plus y squared. So writing that out, we have our tea being equal Thio this is going to be the vector, so just write it that way. She, um sure Duty. My next t squared. This is a parametric equation that we have here. Um and it ultimately allows us to assign these different values. Um, this is going to be zero. We see that X equals t Why equals zero and Z, we know is equal to t duty minus t squared. So with that, we know that there's thedc, Carrabba, Lloyd equation X squared plus y squared equals e. So we're gonna plug in these different values. So busy equals X squared plus y squared. We know that we can plug in this for Z. You can plug in this right here for X, so we'll get CI squared and then we know that why is zero? So it's just gonna be like this right here, then we will add, uh, to our a T Square to both sides. Giving us two t equals to t squared. When we do that, we can actually bring it over to this side. So we'll have to t minus two t squared equals zero a factor attitude t equals one minus g serum. When we do that, we get that t equals zero and t equals one. So with that in mind, what's determinar different values now? So we'll have ex being equal to either zero. Why is always going to be zero, which means we would have a Z value of zero and then the other possibility is that X equals one. Why will still equals zero? But then Z would ultimately equal one, So these would be are two possible points of intersection.