Polar Equation of a Conic

polarcon.html

Exploration

Let the focus F of a conic be at the pole of a polar coordinate system and the directrix D be perpendicular to the polar axis at a distance ρ to the left of the pole. Show that the polar equation of the conic is

where e is the eccentricity of the conic.

Approach

Use the definition of eccentricity e=PF/PD and substitute the expressions for distances. Solve the resulting equations for r.

Initialize

To initialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Use the definition of eccentricity.

Clear[e,PF,PD];
eq1=e==PF/PD

Substitute the distances for the segment lengths.

Clear[r,p,t];
eq2=eq1 /.
    {PF->r, PD->p+r*Cos[t]}

Solve for r.

Solve[eq2,r] //Simplify