An implicit function of two variables

For the given relation $$x^2 + 2y^2 + 3z^2 + xy - z - 9 = 0$$.

• Variant 1

Prove that this relation defines a smooth function $$z = z(x, y)$$, in some neighborhood $$U$$ of the point $$[1, -2]$$ satisfying $$z(1, -2) = 1$$.

• Variant 2

Determine $$\frac{\partial z}{\partial x}$$ and $$\frac{\partial z}{\partial y}$$ in the neighborhood $$U$$.

• Variant 3

Write down the equation of the plane tangent to the function $$z$$ at the point $$[1, -2]$$.