Cônicas e suas propriedades refletoras

Abstract

The objective of this work is to present and demonstrate the reflective properties of the conics, as well as to understand why these geometric figures have fascinated mathematicians since antiquity. Mathematical demonstrations were prioritized with the help of Geometry and Algebra. The use of differential and integral calculus resources was avoided, since this work is aimed at the students of the Middle School, who do not have knowledge of these mathematical resources. In a first moment we analyzed the reasons why these geometric figures receive so little attention in the curriculum of Middle School. It was opportune to analyze a small collection of books recommended by the Ministry of Education (MEC), listed in National Textbook Program (PNLD). It is possible to perceive a very superficial approach to conics, especially hyperbole and ellipse, although the parable in some books is studied in more depth, normaly representing the graphic of a quadratic function. Next we present a brief exposition on the historical origins of the conics, where we highlight the four main protagonists of the theme: Pythagoras, Euclid, Archimedes and Apollonius. We study, separately, each conic from its definition, followed by an algebraic development to find the equation that defines it. We emphasize their reflective properties and how to use them in the creation of technological equipment that helps in the scientific evolution of man. Following are examples of equipment that use technology using the principles of conics. The software Geogebra, Paint.net, Google Sketchup 8.0 and Gimp 2.0, were used as computational tools to elaborate the figures in this work.