5 - ANALYTIC GEOMETRY: THE CONIC SECTIONS

Abstract

This chapter discusses analytic geometry. Analytic geometry enables applying algebraic methods and equations to the solution of problems in geometry and, conversely, to obtain geometric representations of algebraic equations. It is also possible to obtain a formula for the coordinates (x, y) of the midpoint P of the line segment whose endpoints are P1 and P2. Passing lines through P and P2 parallel to the y-axis and a line through P1, parallel to the x-axis results in the similar right triangles P1AP and P1BP2. The conic sections provide an outstanding opportunity to illustrate the double-edged power of analytic geometry. A geometric figure defined as a set of points can often be described analytically by an algebraic equation. If a plane is passed through a cone at various angles, the intersections are called conic sections. In exceptional cases, the intersection of a plane and a cone is a point, a line, or a pair of lines.