Analytic geometry is roughly the same as plane geometry except that in analytic
geometry, figures are studied in the context of the coordinate plane. Instead
of focusing on the congruence of shapes like plane geometry, analytic geometry
deals with the coordinates of shapes and formulas for their graphs in the
coordinate plane. Much of analytic geometry focuses on the conics. A conic
is a two-dimensional figure created by the intersection of a plane and a right
circular cone. All conics can be written in terms of the following equation:
Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0. The four conics we'll explore in this
text are parabolas, ellipses, circles, and hyperbolas. The
equations for each of these conics can be written in a standard form, from
which a lot about the given conic can be told without having to graph it. We'll
study the standard forms and graphs of these four conics, as well as the
polar equations for the conics, which are useful for certain applications.