A Parametric Study of Tangents, Normals & Conjugate Diameters of an Ellipse

This research article delves into the geometric and mathematical significance of the Ellipse; a fundamental conic section defined as the locus of a point such that the ratio of its distance from a fixed point (focus) to a fixed line (directrix) is a constant less than or equal to one. Beyond its classical geometric definition, the ellipse holds substantial importance in astronomy, particularly through Kepler’s laws of planetary motion, where celestial bodies orbit stars in elliptical paths. The study presents a collection of Twenty-nine newly established theorems focusing on the ellipse’s tangent & normal of pair conjugate diameters. Each theorem is rigorously formulated, accompanied by detailed mathematical proofs and illustrative diagrams. Furthermore, the author introduces an additional set of 29 original theorems describing precise mathematical relationships between tangent, normal of pair conjugate diameters of an ellipse, and other key elements of the ellipse. Basic fundamental formulae required for derivations have been given in Preamble & Table-1. These contributions offer valuable insights and serve as significant references for scholars pursuing advanced research in geometry and related disciplines.