Parabola that intersects two lines and matching the slope of the two lines?

Question

If I have two lines with equations;$$x=0$$ $$y=0$$ $$z=t$$ and $$x=t$$ $$y=10$$ $$z=t$$ are there any parabolas that cross through the two lines and in which the parabola matches the slope of the lines at the points of intersection?

Answer 1

No. All of the tangent lines of a parabola lie in a plane (in fact, the entire parabola lies in the same plane). The two lines you gave are of course their own tangent lines, so the parabola has to lie in some translation of the plane spanned by $(0,0,1)$ and $(1,0,1)$ (which is the $xz$-plane), so the the parabola has a constant $y$ value. It cannot be both 10 and 0.