Strange behavior of $B=2$ in the equation $x^2 + 2xy +y^2 = 1$

Question

I Was exploring the behavior of graphs in the form $Ax^2 + Bxy + Cy^2 +Dx +Ey +F$ on DESMOS and $x^2 + xy + y^2 = 1$ makes a fairly simple ellipse, with its major axis along $y =-x$ and centered on the origin.

When I made the coefficient (B) of the 'xy' term 2, $(x^2 + 2xy + y^2 = 1)$, What I got on the graph appeared to be two parallel lines: $y = -x+1$ and $y = -x-1$.

As B closely approached 2, the ellipse began to elongate dramatically; as B exceeded 2 even slightly, the graph started to become a hyperbola.

What property of the coefficients A, B, and C (if that's what it is) is generating these two parallel lines when B = 2 in this particular equation? I'm baffled.

Thank You,

Victor Jaroslaw

Answer 1

$x^2+2xy+y^2=(x+y)^2$ is a square of a linear function. So $(x+y)^2=1$ iff either $x+y=1$ or $x+y=-1$, that is, $y=1-x$ or $y=-1-x$. In general, the behaviour of a curve of the form $$ax^2+bxy+cy^2=1$$ is determined by $D=b^2-4ac$, the discriminant. If $D<0$ the curve is an ellipse, if $D>0$ it is a hyperbola, while if $D=0$ it degenerates into a pair of lines (possibly coincident).