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 two polar equations. The first is given by
For the other polar equation (just reversing the sign on the absolute value above) we have