Example 3.39. Write the equation of the tangent plane and the normal line to the surface $z=2 x^{2}+3 y^{2}$ at the point $M(1,2,14)$.
Solution. The required equations are found using formulas (3.32) and (3.33):

$$
f(x, y)=2 x^{2}+3 y^{2}, f_{x}^{\prime}(M)=4 x=4, f_{y}^{\prime}(M)=6 y=12
$$

The equation of the tangent plane is

$$
z-14=4(x-1)+12(y-2) \Rightarrow 4 x+12 y-z-14=0
$$

The equation of the normal line $\frac{x-1}{4}=\frac{y-2}{12}=\frac{z}{-1}$.