I have a question which asks me to find the equation of a line perpendicular to the ellipse x^2-xy+y^2=3, at the point (-1, 1). I did this by differentiating implicitly, finding the tangent line then the line perpendicular to that. Pretty simple. The equation of the perpendicular line turned out to be f(x)=-x.

The question now is where does this line intersect the ellipse again? I know the answer is (1, -1) by looking at the graph, but I need to justify this mathematically. Anyone know any algebra gymnastics which will lead me to this answer, based only on f(x)=-x, and the point (-1, 1)?