Finding horizontal tangent line using implicit differentation

Question

How do you find the horizontal tangent line from an equation like this: $x^2 +xy +y^2=6$? I've already found the derivative using implicit differentiation and tried solving for 0...but I have two variables.

Answer 1

HINT:

On implicit differentiation, $$2x+x\frac{dy}{dx}+y+2y\frac{dy}{dx}=0$$

$\displaystyle\frac{dy}{dx}$ denotes the tangent line at $(x,y)$

The slope/gradient of horizontal tangent line $=0$

This will give us a relation between $x,y$

Solve for $x,y$ using the given equation of the curve