Tangents [1]

Let $z_1 (x,y) = 2x^2+y^2$ and $z_2 (x,y) = Ax+By-C$ . Since the plane $z_2$ is tangent to the paraboloid $z_1$ at $(1,1,3)$ , we have:

$\begin{cases} \dfrac {\partial z_2}{\partial x} \bigg|_{x=1} = \dfrac {\partial z_1}{\partial x} \bigg|_{x=1} & \implies A = 4x \bigg|_{x=1} = 4 \\ \dfrac {\partial z_2}{\partial y} \bigg|_{y=1} = \dfrac {\partial z_1}{\partial y} \bigg|_{y=1} & \implies B = 2y \bigg|_{y=1} = 2 \\ z_2 (1,1) = z_1 (1,1) & \implies \begin{aligned} A + B - C & = 3 \\ 4+2 - C & = 3 \\ \implies C & = 3 \end{aligned} \end{cases}$

Therefore, $A+B+C = 4+2+3 = \boxed 9$ .