Consider a conic section with focus at the origin, eccentricity and directrix given by

Compute the distance from the focus to the directrix and find a polar equation for . For a hyperbola, give a polar equation for each branch.

First, the distance from any point to the directrix is given by the equation

where is any point on the directrix . In this case we can choose as a point on the directrix and let be the focus. Then the distance from the focus to the directrix is

Since a normal to the directrix is (from the Cartesian equation for the directrix) we have

To obtain the polar equation for the conic section, since the focus is at the origin, we have the conic section is the set of points such that

Letting have polar coordinates and , (i.e., ), and letting be the unit normal we have the polar equation

Is there any way to know which way to calculate the absolute value (ie to take the positive or negative of whatever’s inside the absolute value)? The only thing I can think of is to calculate both versions and see which one implies r < 0

Thanks!