simplest common equation of conics

In the plane, the locus of the points having the ratio of their distancesMathworldPlanetmath from a certain point (the focus) and from a certain line (the directrixPlanetmathPlanetmathPlanetmath) equal to a given constant ε, is a conic section, which is an ellipseMathworldPlanetmath, a parabola ( or a hyperbolaMathworldPlanetmath depending on whether ε is less than, equal to or greater than 1.

For showing this, we choose the y-axis as the directrix and the point  (q, 0)  as the focus.  The locus condition reads then


This is simplified to

(1-ε2)x2-2qx+y2+q2= 0. (1)

If  ε=1,  we obtain the parabola

y2= 2qx-q2.

In the following, we thus assume that  ε1.

Setting  y:=0  in (1) we see that the x-axis cuts the locus in two points with the midpointMathworldPlanetmathPlanetmathPlanetmath of the segment connecting them having the abscissaMathworldPlanetmath


We take this point as the new origin (replacing x by x+x0); then the equation (1) changes to

(1-ε2)x2+y2=ε2q21-ε2. (2)

From this we infer that the locus is

  1. 1.

    in the case  ε<1  an ellipse ( with the semiaxes


    and with eccentricity ε;

  2. 2.

    in the case  ε>1  a hyperbola ( with semiaxes


    and also now with the eccentricity ε.


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 q be the distance of the focus from the corresponding directrix.  Let r and φ be the polar coordinatesMathworldPlanetmath of an arbitrary point of the conic.  Then the locus condition may be expressed as


Solving this equation for the radius r yields the form

r=εq1εcosφ (3)

for the common polar equation of the conic.  The sign alternative () depends on whether the polar axis (φ=0) intersects the directrix or not.

Title simplest common equation of conics
Canonical name SimplestCommonEquationOfConics
Date of creation 2015-03-12 8:24:02
Last modified on 2015-03-12 8:24:02
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Derivation
Classification msc 51N20
Synonym common equation of conics
Related topic ConicSection
Related topic QuadraticCurves
Related topic BodyInCentralForceField