a) Let the curve be parametrized by arclength, and let be the Frenet-Serret frame. Then the plane always passes through a point . Express and differentiate to obtain . So always crosses the origin, which means or . Use the Frenet-Serret equations to show that are constants.
(Actually, the curvature of turns out to be constant and the torsion zero, so is a circle.)
b) Let the coordinates be x,y,z. We easily see that this circle is the intersection of the circular cylinder with the parabolic cylinder .
(...unless i messed up my calcs again)