An *ellipse*, centered on the origin, of major radius and minor radius , which are aligned
along the - and -axes, respectively (see Figure 14), satisfies the following
well-known equation:

Likewise, a parabola which is aligned along the -axis, and passes through
the origin (see Figure 15), satisfies:

Finally, a hyperbola which is aligned along the -axis, and whose
asymptotes intersect at the origin (see Figure 16), satisfies:

It is not clear, at this stage, what the ellipse, the parabola, and the hyperbola
have in common (other than being conic sections). Well, it turns out that what these three curves
have in common is that they can all be represented as the locus of a movable point whose distance from
a fixed point is in a constant ratio to its perpendicular distance to some
fixed straight-line. Let the fixed point (which is termed the *focus*
of the ellipse/parabola/hyperbola) lie at the origin, and let
the fixed line correspond to (with ). Thus, the distance of a general point (, ) (which lies to the right of the line ) from the origin is
, whereas the perpendicular distance of the point from
the line is --see Figure 17.
In polar coordinates, and
.
Hence, the locus of a point for which
and are in a fixed ratio satisfies the following equation:

where .

When written in terms of Cartesian coordinates, (233)
can be rearranged to give

Equation (235) can be recognized as the equation of an

When again written in terms of Cartesian coordinates, Equation (233)
can be rearranged to give

(239) |

Finally, when written in terms of Cartesian coordinates, Equation (233)
can be rearranged to give

(241) | |||

(242) | |||

(243) |

Equation (240) can be recognized as the equation of a

(244) |

In conclusion, Equation (234) is the polar equation of a *general conic
section* which is *confocal with the origin*. For , the conic section
is an *ellipse*. For , the conic section is a *parabola*. Finally, for
, the conic section is a *hyperbola*.