Find the point(s) where the following plane and curve intersect:
2x+3y-9z=0
r(t)= <3cost, 3sint, cost> for [0,2pi]
Just plug into the equation for the plane and solve for t!
$\displaystyle 6\cos(t)+9\sin(t)-9\cos(t)=0 \iff 9\sin(t)=3\cos(t) \iff \tan(t)=\frac{1}{3}$
Now taking the arctan of both sides gives
$\displaystyle t=\tan^{-1}\left( \frac{1}{3}\right)+\pi n, n \in \mathbb{Z}$
Now just restric n to get the solutions in the needed interval.