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In this paper, we bring forth several new general formulae in the classic study of conics in the Analytic Geometry: the coordinates of all vertices and focal points of arbitrary parabolas, ellipses, and hyperbolas; lengths for all latera recta from any non-degenerate conic section; equations describing straight lines whose limited-slope contents stand on exactly equal footing as focal axes, latera recta, and directrices from every non-degenerate conic section; and, respectively, these ones characterizing asymptotes for each non-degenerate hyperbola. All these general results work regardless of whether the conics in question are rotated or not on the Cartesian plane, because all of them depend only on the coefficients of the general conic equation, making the rotation angle irrelevant for the analysis of conic sections.

We present a general formula that transforms any conic of the form Ax2+Bxy+Cy2+Dx+Ey+F=0, with B≠0, into A′(x′)2+C′(y′)2+D′x′+E′y′+F=0, without requiring the rotation angle θ. This directly eliminates the cross term xy, simplifying the rotated conics analysis. As consequences, we obtain new formulae that remove both rotations and translations, a novel proof of the discriminant criterion, improved expressions for eccentricity, and a detailed taxonomy of all loci described by the general conic equation.