Determining the type of conic and drawing a graph of it

Question

What type of conic is given by the equation:

$x^2+2xy+y^2-1=0$

And draw a graph of the conic.

My method was to compute the intersection of the conic with the line at infinity, and got $x=-y$ so we have one point of intersection and hence it is a parabola? I noticed that the symmetric matrix for this conic is singular so this means the conic is degenerate. So is it still a parabola? Would the graph just be a line?

Answer 1

The equation factors as

$$(x+y)^2-1=(x+y+1)(x+y-1)=0$$ and describes a pair of parallel straight lines.

Answer 2

$${ \left( x+y \right) }^{ 2 }=1\\ y=\pm 1-x$$ they are parallel lines