Answer
$$\left( {0,3,12} \right)$$
Work Step by Step
$$\eqalign{
& z = 3 - {x^2} - {y^2} + 6y \cr
& z - 3 + {x^2} + {y^2} - 6y = 0 \cr
& {\text{Consider}} \cr
& F\left( {x,y,z} \right) = z - 3 + {x^2} + {y^2} - 6y \cr
& {\text{Calculating the partial derivatives}} \cr
& {F_x}\left( {x,y,z} \right) = 2x \cr
& {F_y}\left( {x,y,z} \right) = 2y - 6 \cr
& {F_z}\left( {x,y,z} \right) = 1 \cr
& {\text{Find the gradient }}\nabla F\left( {x,y,z} \right) \cr
& \nabla F\left( {x,y,z} \right) = 2x{\bf{i}} + \left( {2y - 6} \right){\bf{j}} + {\bf{k}} \cr
& {\text{Find the point}}\left( {\text{s}} \right){\text{ on the surface at which the tangent plane }} \cr
& {\text{is horizontal}}. \cr
& 2x = 0 \to x = 0 \cr
& 2y - 6 = 0 \to y = 3 \cr
& z = 3 - {x^2} - {y^2} + 6y \cr
& z = 3 - {\left( 0 \right)^2} - {\left( 3 \right)^2} + 6\left( 3 \right) \cr
& z = 12 \cr
& {\text{The point is}} \cr
& \left( {0,3,12} \right) \cr} $$