The conic section with equation x^2-y^2 = 0 is degenerate as its equation can be written as (x-y)(x+y)= 0, and corresponds to two intersecting lines forming an "X".
The pencil of ellipses of equations ax^2+b(y^2-1)=0 degenerates, for a=0, b=1, into two parallel lines and, for a=1, b=0, into a double line.