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Mathematics
Volume 14
Issue 1
10.3390/math14010138
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Article

A Unified General Theory of Conic Sections via the Conic Radical

by
Mauricio Chávez-Pichardo
1,
José Daniel López-Barrientos
2,* and
Saúl Perea-Flores
3
1
TecNM–Tecnológico de Estudios Superiores del Oriente del Estado de México, División de Estudios de Posgrado e Investigación y División de Ingeniería en Energías Renovables, La Paz 56400, Mexico
2
Facultad de Ciencias Actuariales, Universidad Anáhuac México, Huixquilucan de Degollado 52786, Mexico
3
PNC Financial Services Group, Texas Division, Dallas, TX 75206, USA
*
Author to whom correspondence should be addressed.
Mathematics 2026, 14(1), 138; https://doi.org/10.3390/math14010138 (registering DOI)
Submission received: 6 October 2025 / Revised: 26 November 2025 / Accepted: 24 December 2025 / Published: 29 December 2025
(This article belongs to the Section B: Geometry and Topology)
Versions Notes

Abstract

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.
Keywords: analytic geometry; conic sections; general conic equation; general second-degree equation; rotated conic sections; translation and rotation of the Cartesian axes; general theorem of conic sections analytic geometry; conic sections; general conic equation; general second-degree equation; rotated conic sections; translation and rotation of the Cartesian axes; general theorem of conic sections

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MDPI and ACS Style

Chávez-Pichardo, M.; López-Barrientos, J.D.; Perea-Flores, S. A Unified General Theory of Conic Sections via the Conic Radical. Mathematics 2026, 14, 138. https://doi.org/10.3390/math14010138

AMA Style

Chávez-Pichardo M, López-Barrientos JD, Perea-Flores S. A Unified General Theory of Conic Sections via the Conic Radical. Mathematics. 2026; 14(1):138. https://doi.org/10.3390/math14010138

Chicago/Turabian Style

Chávez-Pichardo, Mauricio, José Daniel López-Barrientos, and Saúl Perea-Flores. 2026. "A Unified General Theory of Conic Sections via the Conic Radical" Mathematics 14, no. 1: 138. https://doi.org/10.3390/math14010138

APA Style

Chávez-Pichardo, M., López-Barrientos, J. D., & Perea-Flores, S. (2026). A Unified General Theory of Conic Sections via the Conic Radical. Mathematics, 14(1), 138. https://doi.org/10.3390/math14010138

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