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Boundless Algebra
Complex Numbers and Polar Coordinates
The Polar Coordinate System
Algebra Textbooks Boundless Algebra Complex Numbers and Polar Coordinates The Polar Coordinate System
Algebra Textbooks Boundless Algebra Complex Numbers and Polar Coordinates
Algebra Textbooks Boundless Algebra
Algebra Textbooks
Algebra
Concept Version 5
Created by Boundless

Conics in Polar Coordinates

Polar coordinates allow conic sections to be expressed in an elegant way.

Learning Objective

  • Recognize the equations for different conic sections in polar coordinates


Key Points

    • Conic sections have several key features which define their polar equation; foci, eccentricity, and a directrix.
    • All conic sections have the same basic equation in polar coordinates, which demonstrates a connection between all of them.

Terms

  • eccentricity

    A measure of deviation from a prescribed curve. 

  • directrix

    A fixed line used to described a curve.


Full Text

Defining a Conic

Previously, we learned how a parabola is defined by the focus (a fixed point) and the directrix (a fixed line). 

Parts of a Parabola

Consider the parabola $x=2+y^2$. Any conic may be determined by three characteristics: a single focus, a fixed line called the directrix, and the ratio of the distances of each to a point on the graph.  

We can define any conic in the polar coordinate system in terms of a fixed point, the focus $P(r,θ)$ at the pole, and a line, the directrix, which is perpendicular to the polar axis.

For a conic with eccentricity $e$,

  1. If $0≤e<1$, the conic is an ellipse.
  2. If $e=1$, the conic is a parabola.
  3. If $e>1$, the conic is an hyperbola.

With this definition, we may now define a conic in terms of the directrix: $x=±p$, the eccentricity $e$, and the angle $\theta$. Thus, each conic may be written as a polar equation in terms of $r$ and $\theta$.

For a conic with a focus at the origin, if the directrix is $x=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:  

$\displaystyle r=\frac{e\cdot p}{1\: \pm\: e\cdot\cos\theta}$

For a conic with a focus at the origin, if the directrix is $y=±p$, where $p$ is a positive real number, and the eccentricity is a positive real number $e$, the conic has a polar equation:  

$\displaystyle r=\frac{e\cdot p}{1\: \pm\: e\cdot\sin\theta}$

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