Explanation of graphical mathematical anomaly (for me, anyways)

Question

I was working on some competition stuff when I came across the equation $y^2+2xy-x^2 = 0$, and the thing that surprised me was, when I graphed it, I got these two perpendicular lines at the origin, each slightly rotated. I cannot explain this in any way and have been searching online for hours, only to recognize that it is some conic section (I think). All help would be appreciated.

Answer 1

The given relation is indeed a conic section, although degenerate, corresponding to the intersection of a double cone by a plane that passes through the cone's axis.

To see why the given relation defines two lines, it simply suffices to solve for one of the variables in terms of the other, either through completing the square or explicitly using the quadratic formula. Using the first method yields $$y^2 + 2xy + x^2 = 2x^2$$ from which we obtain $$(x+y)^2 = 2x^2,$$ hence $$y = -x \pm \sqrt{2x^2} = x(-1 \pm \sqrt{2}).$$ Therefore, the relation is equivalent to the union of the two lines $$\begin{align*} y &= (\sqrt{2}-1)x, \\ y &= -(\sqrt{2} + 1)x. \end{align*}$$ Since their slopes have product $-(\sqrt{2}+1)(\sqrt{2}-1) = -1$, it also follows that these lines are mutually perpendicular.

Answer 2

Hint:

$y^2+2xy-x^2=(y+(1-\sqrt{2})x)(y+(1+\sqrt{2})x)$