Polar equations for conic sections

We will work with conic sections with a focus at the origin.

Polar equations of conic sections: If the directrix is a distance $d$ away, then the polar form of a conic section with eccentricity $e$ is $$r(\theta) = \frac{ed}{1-e \cos(\theta-\theta_0)},$$ where the constant $\theta_0$ depends on the direction of the directrix.

This formula applies to all conic sections. The only difference between the equation of an ellipse and the equation of a parabola and the equation of a hyperbola is the value of the eccentricity $e$. There are four important special cases:

If the directrix is the line $x=d$, then we have $\displaystyle{r = \frac{ed}{1+e\cos(\theta)}.}$ If the directrix is the line $x=-d$, then we have $\displaystyle{r = \frac{ed}{1-e\cos(\theta)}.}$ If the directrix is the line $y=d$, then we have $\displaystyle{r = \frac{ed}{1+e\sin(\theta)}.}$ If the directrix is the line $y=-d$, then we have $\displaystyle{r = \frac{ed}{1-e\sin(\theta)}.}$

In all cases, the curve opens out in the direction opposite to the directrix.