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:  

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