For the curve described by this expression answer the following questions:
$$x^2 + xy + y^2 = 7$$
Find the equation(s) of all lines tangent to the curve at $x = -1$.
Give line in slope-intercept form. If there is more than one line, please indicate to which point the line corresponds.
HINT
To begin with, start by determining the points of the ellipse which have $x = -1$ as its abscissa: \begin{align*} (-1)^{2} + (-1)y + y^{2} = 7 \Longrightarrow y^{2} - y - 6 = 0 \Longleftrightarrow (y = -2)\vee(y = 3) \end{align*}
Then you can make use of implicit differentiation to obtain the relation among $x,y$ and $y'$: \begin{align*} x^{2} + xy + y^{2} = 7 \Longrightarrow 2x + y + xy' + 2yy' = 0 \end{align*}
Now plugin the pairs of values $(-1,-2)$ and $(-1,3)$ with the purpose of finding the corresponding slopes.
Can you take it from here?