Question

At what points does the curve r(t) = ti + (3t − t2)k intersect the paraboloid z = x2 + y2? (if an answer does not exist, enter dne.) (x, y, z) = (smaller t-value) (x, y, z) = (larger t-value)

Answer 1

$\mathbf r(t)=x(t)\,\mathbf i+y(t)\,\mathbf j+z(t)\,\mathbf k=t\,\mathbf i+(3t-t^2)\,\mathbf k$

Along the curve $\mathbf r(t)$, we have $y(t)=0$, so the equation of the paraboloid reduces to

$z=x^2\iff3t-t^2=t^2\implies2t^2-3t=t(2t-3)=0\implies t=0\text{ and }t=\dfrac32$