# Analytic Geometry

## Overview

### Description

Analytic geometry includes the study of conic sections in the coordinate plane. When a cone or cones intersect with a plane, the cross section is called a conic section. A conic section can be a circle, an ellipse, a hyperbola, or a parabola. The equation of a conic section can be written in general form, $Ax^2+By^2+Cx+Dy+E=0$, in which the coefficients of the variables determine the type of conic section that is formed. Each type of conic section also has a standard form of its equation. The standard form is used to identify information that is used to graph each type of conic section. It may be necessary to complete the square to rewrite an equation that is not given in standard form in order to graph the equation.

### At A Glance

• The intersection of a double cone and a plane can form different conic sections.
• A circle in the coordinate plane can be identified by its center and radius and can be defined by an equation.
• A circle with a given equation can be graphed in the coordinate plane by determining the center and radius.
• An ellipse is a conic section defined by two fixed points. The sum of the distances between these two fixed points and any point on the ellipse is constant.
• An ellipse with a given equation can be graphed in the coordinate plane by locating the center, vertices, and co-vertices.
• A hyperbola is a conic section defined by two fixed points. The difference of the distances between these two fixed points and any point on the hyperbola is constant.
• A hyperbola with a given equation can be graphed in the coordinate plane by locating the center and vertices and sketching its asymptotes and fundamental rectangle.
• A parabola is a conic section defined in terms of distance to a fixed point and a fixed line.
• A parabola with a given equation can be graphed in the coordinate plane by locating the vertex, focus, and directrix.