Key Concepts

• Any conic may be determined by a single focus, the corresponding eccentricity, and the directrix. We can also define a conic in terms of a fixed point, the focus [latex]P\left(r,\theta \right)[/latex] at the pole, and a line, the directrix, which is perpendicular to the polar axis.
• A conic is the set of all points [latex]e=\frac{PF}{PD}[/latex], where eccentricity [latex]e[/latex] is a positive real number. Each conic may be written in terms of its polar equation.
• The polar equations of conics can be graphed.
• Conics can be defined in terms of a focus, a directrix, and eccentricity.
• We can use the identities [latex]r=\sqrt{{x}^{2}+{y}^{2}},x=r\text{ }\cos \text{ }\theta[/latex], and [latex]y=r\text{ }\sin \text{ }\theta[/latex] to convert the equation for a conic from polar to rectangular form.

Glossary

eccentricity

the ratio of the distances from a point [latex]P[/latex] on the graph to the focus [latex]F[/latex] and to the directrix [latex]D[/latex] represented by [latex]e=\frac{PF}{PD}[/latex], where [latex]e[/latex] is a positive real number

polar equation

an equation of a curve in polar coordinates [latex]r[/latex] and [latex]\theta[/latex]