<s>[INST] $Ax^2+Bxy+Cy^2=1$ with $A=C=1$ and $B=2$ should be a parabola (because $B^2 = 4 AC$). Instead, it represents parallel lines. What went wrong? My calculus book says

The equation $Ax^2+Bxy+Cy^2=1$ produces a hyperbola if $B^2>4AC$ and
  an ellipse if $B^2<4AC$. A parabola has $B^2=4AC$.

When I set $B=2,A=C=1$, the equation becomes $x^2+2xy+y^2=1$, which is equivalent to $(x+y)^2=1$ or $y=-x\pm1$. This is not a parabola but two parallel lines. What went wrong?
 [/INST] You are right: a nondegenerate parabola doesn't admit an equation of the form $Ax^2+2Bxy+Cy^2=1$.
Indeed, if $B^2=AC\ne0$, we can multiply all coefficients by $A$, finding
$$
A^2x^2+2ABxy+B^2y^2=B
$$
that becomes $(Ax+By)^2=B$. If $B>0$ this factors as
$$
(Ax+By-\sqrt{B})(Ax+By+\sqrt{B})=0
$$
which is a pair of parallel lines.
If $B<0$, the factorization is over the complex numbers and we get no real point.
If $B=0$, then either $A=0$ or $C=0$, so the equation becomes either $Cy^2=1$ or $Ax^2=1$, again a situation as before.
Only nondegenerate conics with a center admit an equation of the given form, which is obtained by translating the center of the conic in the origin.
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