key term - Polar equation

Definition

A polar equation is a mathematical expression that defines a relationship between the radius $r$ and the angle $\theta$ of a point in the polar coordinate system. It is commonly used to describe conic sections and other geometric shapes.

5 Must Know Facts For Your Next Test

1. Polar equations use $r$ (radius) and $\theta$ (angle) instead of $x$ and $y$ coordinates.
2. Common forms of polar equations for conic sections include $r = \frac{ed}{1 + e \cos(\theta)}$ for ellipses, hyperbolas, and parabolas.
3. The eccentricity $e$ determines the type of conic section: ellipse $(0 < e < 1)$, parabola $(e = 1)$, or hyperbola $(e > 1)$.
4. Converting between Cartesian and polar coordinates involves the formulas: $x = r \cos(\theta)$ and $y = r \sin(\theta)$.
5. Graphs of polar equations can exhibit symmetry about the pole (origin), the line $\theta = \frac{\pi}{2}$, or the polar axis.

Review Questions

Related terms

Polar Coordinates:

A coordinate system where each point on a plane is determined by an angle and a distance from a reference point called the pole.

Eccentricity:

A parameter that determines the shape of a conic section; calculated as $e = \frac{c}{a}$ where $c$ is the distance from center to focus, and $a$ is the distance from center to vertex.

$r = \frac{ed}{1 + e \cos(\theta)}$: The standard form of a polar equation representing various conic sections depending on the value of eccentricity $e$. For instance, it represents an ellipse when $0 < e < 1$, a parabola when $e = 1$, and a hyperbola when $e > 1$.