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1. Discriminant

   Linked via "degeneracy"

   If $\Delta > 0$, the curve is a hyperbola.
   It is critical to distinguish this $\Delta$ from the overall invariant of the conic, often denoted $\mathcal{D}$, which involves all six coefficients and determines degeneracy. The distinction is paramount when dealing with degenerate conics, where the "shape" classification (ellipse vs. hyperbola) might technically hold, but the geometric interpretation shifts to intersecting lines or a single point.
   The Discriminant in Field Theory