Topic: Parametric, Polar, and Conic Curves

Subtopic: Polar Equations of Conic Sections

**Overview**

Start by reviewing conic sections from a pre-calculus course. See Lesson Notes - College Algebra: Introduction to Conics | Parabolas | Ellipses | Hyperbolas | General & Degenerate Conics but ignore any "text notes". Work conics that are centered at the origin as well as those translated away from the origin.

Our main goal in this lesson is to concentrate performing calculus on the conic sections. Many calculus operations are simplified by converting the equations of the conics from rectangular form to polar form. This transformation is accomplished through an ingenious formula. We can then apply calculus to the polar form to find the area inside a bounded conic, the arc length of the conic, etc. Study conics that have horizontal or vertical axes as well as those that have a rotated axis.

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 10.4.1 Understand the rectangular equations and graphs of conic sections including those with centers translated away from the origin
- 10.4.2 Use the formula `r = (e*d)/(1+-costheta)` or `r = (e*d)/(1+-sintheta)` to convert equations of conics from rectangular to polar form (formulae in box, below)
- 10.4.3 Identify a conic from its polar form
- 10.4.4 Understand the eccentricity of the conic sections and identify a conic given its eccentricity
- 10.4.5 Graph a polar equation of a conic manually and electronically
- 10.4.6 Find the polar equation of a conic given its graph or information about its graph
- 10.4.7 Understand some applications of conics in polar form including Kepler's Laws

**Terminology & Formulae**

Terms you should be able to define: focal point, major and minor axes, directrix, pole, eccentricity "e", Kepler and his laws

**Mini-Lectures and Examples**

STUDY: Polar Equations of Conic Sections

**Supplemental Resources (recommended)**

As a review of Conic Sections from Pre-Calculus, you may want to download one of the following formula sheets, whichever works best for you:

- Conic Sections Formula Sheet by Jean Adams, Flamingo Math
- Conic Sections Formulas and Summary Chart by Prof. Louise Hoover, retired, Clark College
- Conic Sections Formulas by Diane Krasnewich, TTDK.com

Watch The Organic Chemistry Tutor's Calculus II video Eccentricity of an Ellipse and/or explore the connection between eccentricity and conics via the interactive graph at Math Is Fun: Eccentricity.

**Supplemental Resources (optional)**

These are both quite good reads. The first reviews concepts from Pre-Calculus and the second covers Calculus concepts.

Lesson: Conic Sections, Dale Hoffman's Contemporary Calculus

Lesson: Properties of the Conic Sections, Dale Hoffman's Contemporary Calculus

This video was recommended by a student who found it helped pull some Pre-Calculus concepts together with the new way of looking at conics via eccentricity: Thinkwell Vids: The Eccentricity of an Ellipse.

rev. 2021-03-06