In the plane, the locus of the points having the ratio of their distances from a certain point (the focus) and from a certain line (the directrix) equal to a given constant , is a conic section, which is an ellipse, a parabola or a hyperbola depending on whether is less than, equal to or greater than 1.
For showing this, we choose the -axis as the directrix and the point as the focus. The locus condition reads then
This is simplified to
If , we obtain the parabola
In the following, we thus assume that .
We take this point as the new origin (replacing by ); then the equation (1) changes to
From this we infer that the locus is
Place the origin into a focus of a conic section (and in the cases of ellipse and hyperbola, the abscissa axis through the other focus). As before, let be the distance of the focus from the corresponding directrix. Let and be the polar coordinates of an arbitrary point of the conic. Then the locus condition may be expressed as
Solving this equation for the polar radius yields the form
for the common polar equation of the conic. The sign alternative () depends on whether the polar axis () intersects the directrix or not.