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Chapter 8: Analytic Geometry in Two and Three Dimensions
Most of this chapter is devoted to the analytic geometry of conic sections, a topic which has diminished somewhat in emphasis since the introduction of so many other topics into the precalculus curriculum. Additionally, a few of the more obvious extensions of two-dimensional coordinate geometry to three-dimensional coordinate geometry are gathered together into the last section. It is hard to strike a happy compromise between "all" and "nothing" when it comes to teaching the conic sections. We have chosen to highlight the geometry of the curves, particularly the reflective properties, which have important applications in the real world. This requires knowing about foci and how to find them, which requires a pretty close look at everything that ties the algebra and the geometry of conics together. The immediate value of this chapter for a first-year calculus course is rather slight, but the material is mandated for precalculus in some state guidelines.



Section 8.1 Conic Sections and Parabolas
**Objectives**

You will be able to describe the geometry of a parabola (vertex, focus, directrix) when given its quadratic equation and will be able to construct the quadratic equation when given sufficient geometric information. You will be able to model and solve real-world problems involving parabolas. **Key Ideas**
 * Chord (of a parabola) || Focus (of a parabola) ||
 * Conic section || Nappe of a cone ||
 * Degenerate conic section || Parabaloid of revolution ||
 * Directrix (of a parabola) || Parabola ||
 * Focal length and focal width || Standard form (of a parabola) ||

Section 8.2 Ellipses
**Objectives**

You will be able to describe the geometry of an ellipse (vertices, foci, major and minor axes) when given its quadratic equation and will be able to construct the quadratic equation when given sufficient geometric information. You will be able to model and solve real-world problems involving ellipses. **Key Ideas**
 * Chord (of an ellipse) || Foci (of an ellipse) ||
 * Eccentricity (of an ellipse) || Major axis ||
 * Ellipse || Minor axis ||
 * Ellipsoid of revolution || Vertices (of an ellipse) ||

Section 8.3 Hyperbolas
**Objectives**

You will be able to describe the geometry of a hyperbola (vertices, foci, transverse and conjugate axes, and asymptotes) when given its quadratic equation and will be able to construct the quadratic equation when given sufficient geometric information. You will be able to model and solve real-world problems involving hyperbolas. **Key Ideas**
 * Chord (of a hyperbola) || Hyperbola ||
 * Conjugate axis || Hyperboloid of revolution ||
 * Eccentricity (of a hyperbola) || Transverse axis ||
 * Foci (of a hyperbola) || Vertices (of a hyperbola) ||

Section 8.4 Translation and Rotation of Axes
**Objectives**

You will be able to rotate coordinate axes in order to graph conic sections that are neither horizontal nor vertical. **Key Ideas**
 * Cross-product term || Rotation formulas ||
 * Discriminant || Rotation of axes ||
 * Invariant under rotation || Translation of axes ||

Section 8.5 Polar Equations of Conics
**Objectives**

You will understand the general focus-directrix definition of conic sections and will be able to write equations of conic sections in polar form. **Key Ideas**
 * Directrix (of a general conic) || Focus (of a general conic) ||
 * Eccentricity (of a general conic) || Polar equation of a conic ||
 * Focal axis (of a general conic) || Vertex (of a general conic) ||

=__Section 8.6 Three-Dimensional Cartesian Coordinate System__= **Objectives**

You will be able to extend two-dimensional formulas from vectors and coordinate geometry to the corresponding formulas for three-dimensions. **Key Ideas**
 * Center (of a sphere) || Radius (of a sphere) ||
 * Octants || Right-handed coordinate frame ||
 * Plane (in Cartesian space) || Sphere ||
 * Quadric surface || Standard unit vectors **i**, **j**, **k** ||