Relações entre grandezas geométricas na elipse: um estudo a partir de uma equação particular

Abstract

Conics are curves studied since ancient times by several mathematicians, the most promi nent of whom was Apollonius of Perga. In the work Treatise on Conic Sections, his definition of an ellipse, when translated into current mathematical language, results in the equation. From this specific equation, we asked ourselves how the different quantities in the ellipse are algebraically related. To answer this question, we set out the following objectives: investigate ways of representing these curves, look for relationships between parameters and quantities, algebraically express the relationships found, analyze the properties and behavior of these relationships and use the dynamic geometry software GeoGebra to represent them graphically. We list 14 elements of the ellipse, described using three different representations: canonical, parametric and polar forms. An applet containing the ellipse in canonical form, its elements and, as an exam ple, one of the relationships found, was built and made available in the GeoGebra online repository. In total, 11 relations of the latus rectum were found and analyzed.