Implicit function theorem - Tangent planes to an Ellipsoid

Question

I have an ellipsoid, whose equation is $ \frac{x^2}{3}+\frac{y^2}{9}+\frac{z^2}{4} = 1$ and a plane whose equation is $ x + y + z = 0$. The goal of this question is to find the tangent planes to the Ellipsoid which are parallel to the plane.

First of all, I do understand that we will get the equations of 2 planes, due to the symmetry of the ellipsoid.

Then, I also know that if we want to find parallel planes, they need to have the same normal vector, $(1,1,1)$. So, I know that the equation of the planes will be somthing like this: $x+y+z+d=0$.

It asks to solve this problem by the implicit function theorem, but despite knowing that with this theorem we can find the normal vector to the surface, I don't know how to proceed.

What shall I do next?

EDIT: Thank you José Carlos for your comment, the equation that I provided was not complete. It is now.

Answer 1

Hint:

The normal vector to the plane, $(1,1,1)$ must be parallel to the gradient vector, $(\frac{2x}3,\frac{2y}9,\frac{2z}4)$. This forms a system of two equations in three unknowns, i.e. of a straight line. It intersects the ellipsoid twice.

Answer 2

@Yves, sorry there's something I am not understanding.

If the normal vector is parallel to the gradient, (and I understand that it is), we conclude that:

$1*k = \frac{2x}{3}$

$1*k = \frac{2y}{9}$

$1*k = \frac{2z}{4}$

And from now on, how do we get a system of 2 equations and 3 variables?