Q.$x^2+y^2+x-y=0$

Two equivalent routes to the same four curves. Algebraically (Definition 5.2), a conic is the locus with $SP=e\cdot PM$ for a fixed focus $S$, directrix, and eccentricity $e$ — expanding this always produces the general second-degree equation $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$. Geometrically, the same curves come from slicing a double-napped cone with a plane: perpendicular to the axis gives a circle; tilted (one nappe only) gives an ellipse; parallel to a generator gives a parabola; parallel to the axis (both nappes) gives a hyperbola — hence the name conic section.

Degenerate forms, when the cutting plane passes through the cone's vertex: a single point ($A=C$, $B=D=E=0$, $F=0$: $x^2+y^2=0$); a line or pair of parallel lines ($A=B=C=0$, the degenerate parabola); a pair of intersecting lines ($A=-C$, rest zero: $x^2-y^2=0$, the degenerate hyperbola).

Identification checklist (axis-aligned case, $B=0$ — the version actually used to solve problems):