key term - Polar form of a conic

Definition

The polar form of a conic is an equation representing conic sections (ellipse, parabola, hyperbola) using polar coordinates $(r, \theta)$. It often involves parameters like the eccentricity $e$ and the directrix.

5 Must Know Facts For Your Next Test

1. The general polar form of a conic is given by $r = \frac{ed}{1 + e\cos(\theta - \theta_0)}$, where $e$ is the eccentricity and $d$ is the distance to the directrix.
2. Eccentricity ($e$) determines the type of conic: $e < 1$ for ellipse, $e = 1$ for parabola, and $e > 1$ for hyperbola.
3. In polar coordinates, one focus of the conic section is placed at the pole (origin).
4. For different values of $\theta$, the equation describes all points $(r, \theta)$ that form the conic.
5. Converting between Cartesian and polar forms requires knowledge of relationships like $x = r\cos(\theta)$ and $y = r\sin(\theta)$.

Review Questions

Related terms

Eccentricity:

A parameter that determines how much a conic section deviates from being circular. It distinguishes between ellipses, parabolas, and hyperbolas.

Directrix:

A fixed line used in describing a curve or surface. In the context of conics in polar form, it helps define their shape along with eccentricity.

Focus (Foci): Points that help define a conic section. For an ellipse or hyperbola in polar coordinates, one focus is at the origin.