Polar Form of Conic Sections - Part 1

In part 1, we derive the equation for the polar form of conic sections. This is easily done with the geometric definition of conic sections where the locus of a point P moves in a plane, such that the ratio of the distance from it to a fixed point F (the focus), to the perpendicular distance from it to a straight line called the directrix is a constant called the eccentricity.

If we give P the polar coordinates of (r,θ), and set the pole at the focus F, we can derive the polar form from PF/PM = e.

\begin{align*} PF&= ePM \\ \Rightarrow r&=e\left ( d-r\cos\theta \right ) \\ \Rightarrow r+er\cos\theta &= ed \\ \Rightarrow r &= \frac{ed}{1+e\cos\theta} \end{align*}