The question is: Let $a$ denote a positive constant. Let $L$ denote the line passing through the two points $(6, 5, 4)$ and $(1, 2, 3)$. Let $S$ denote the surface of the elliptic paraboloid $z = a + 2x^2 + y^2.$ If $L$ is tangent to the surface $S$, find the value of $a$.
Does the approach involves finding the tangent plane and check that the plane contains the line?? I am quite confused.
Hint. The surface is convex. Thus it intersects the line in at most two points. If it is tangent to the line, it must intersect it in only one point. Hence if the line is written as $(x,y,z)=(1+5t,2+3t,3+t),$ where $t\in \mathrm R,$ then we want a value of $a$ for which the equation $59t^2+31t+3+a=0$ has only one solution. That is, the discriminant $$31^2-4(59)(3+a)$$ must vanish. Can you complete it now?