I need to verify if what I am doing is correct here. This is my question:
Write down the equations of tangent lines to the curve of the implicit function $x^2+2x+2y^2−4y=5$ that are normal to the line $y=x+12$.
I don't know why I am struggling with this so much. If the equation of the tangent line is normal to the line $y=x+12$, that would mean that the slope of the normal line is $m=-1$. In that case, if I implicitly differentiate, I get:
$$2x+2+4yy'-4y'=0$$
That would mean I have:
$$2x+2-4y+4=0$$
$$2x-4y+6=0$$
How do I find the points of tangency though? I feel like I am missing very stupid and obvious, but I am not sure.