Geometry: polar equation of conics

Question

In finding polar equations of conics we take focus as the origin but ellipse has 2 focci so which focus is taken as the origin?
Or If if focus is not taken on origin then how to find the polar equation of ellipse?

Answer 1

In the simplest polar equation for the ellipse, take one focus at the origin and the second focus as $r=b,\theta=0$ with$b>0$. Thus the second focus would lie along the positive $x$ axis in rectangular coordinates or the ray $\theta=0$ in polar coordinates.

Let $O$ be the origin, $A$ be the second focus at $r=b>0,\theta=0$ and $P$ be a point on the ellipse. Defining $e$ as the eccentricity of the ellipse we then have from the Law of Cosines on $\triangle OAP$:

$|AP|^2=|OA|^2+|OP|^2-2|OA||OP|\cos \theta$

$((b/e)-r)^2=b^2+r^2-2br\cos \theta$

We expand the left side and note that the quadratic terms in $r$ cancel out leading to an equation that is linear in $r$:

$(b/e)^2-b^2=2br((1/e)-\cos \theta)$

Solving for $r$:

$r=\frac{b(1-e^2)/2e}{1-e \cos \theta}=\frac{a}{1-e \cos \theta}$

where $a$ is the semilatus rectum of the ellipse.