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Mathematics LibreTexts

12: Analytic Geometry

In this chapter, we will investigate the two-dimensional figures that are formed when a right circular cone is intersected by a plane. We will begin by studying each of three figures created in this manner. We will develop defining equations for each figure and then learn how to use these equations to solve a variety of problems. Conic sections are formed when a plane intersects two right circular cones aligned tip to tip and extending infinitely far in opposite directions, which we also call a cone. The way in which we slice the cone will determine the type of conic section formed at the intersection. A circle is formed by slicing a cone with a plane perpendicular to the axis of symmetry of the cone. An ellipse is formed by slicing a single cone with a slanted plane not perpendicular to the axis of symmetry. 

Thumbnail: Conic sections can also be described by a set of points in the coordinate plane. This section focuses on the four variations of the standard form of the equation for the ellipse. An ellipse is the set of all points (x,y)(x,y) in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus(plural: foci).​​​​