765. Find the equations of the tangent plane and normal to the elliptic paraboloid $z=2 x^{2}+y^{2}$ at point A $(1, -1, 3)$.
Solution. Transform the surface equation to the form $2 x^{2}+$ $+y^{2}-z=0$ and, denoting its left side by $F(x, y, z)$, find the partial derivatives $F_{x}^{\prime}=4 x, F_{y}^{\prime}=2 y, F_{z}^{\prime}=-1$, calculate their numerical values at the given point $F_{x}^{\prime}(A)=4, F_{y}^{\prime}(A)=-2$, $F_{2}^{\prime}(A)=-1$ and, substituting into the general equations (I) " (II), we get:

the equation of the tangent plane $4(x-1)-2(x+1)-(z-3)=0$ or $4 x-2 y-z-3=0$;

the equations of the normal $\frac{x-1}{4}=\frac{y+1}{-2}=\frac{z-3}{-1}$.