Question

At what points does the curve r(t) = ti (6t − t2)k intersect the paraboloid z = x2 y2? (if an answer does not exist, enter dne.)

Answer 1

$\mathbf r(t)=t\mathbf i+(6t-t^2)\,\mathbf k=x(t)\mathbf i+y(t)\mathbf j+z(t)\mathbf k$

$\implies z=x^2y^2\iff 6t-t^2=t\times0=0\impiles t(6-t)=0\implies t=0,6$

At $t=0$, you have $\mathbf r(0)=0\mathbf i+0\mathbf j+0\mathbf k$.

At $t=6$, you have $\mathbf r(6)=6\mathbf i+0\mathbf j+0\mathbf k$.