I am trying to solve this question, I understand the logic behind it which I will try to explain after the statement of the question. The question reads:
Let $S$ be an oriented regular surface that is tangent to the plane along a regular curve $\alpha$. Show that the points on $\alpha$ are parabolic or planar points of $S$.
Now considering the Normal Vector field $N$ along $\alpha$, we can see that this normal vector field is a normal vector field along the surface as well.
The plane has the same normal vector plane, and has a Gaussian Curvature of $0$. Thus because the surface has the same normal vector plane and is tangent to the plane there, shouldn't the Gaussian Curvature there also give $0$?
Is my logic sound or is there another way to prove this?