Exploring Analyic Geometry with Mathematica®

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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

"polarcon_1.gif"

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.

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Solution

Use the definition of eccentricity.

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

"polarcon_2.gif"

Substitute the distances for the segment lengths.

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

"polarcon_3.gif"

Solve for r.

Solve[eq2,r] //Simplify

"polarcon_4.gif"


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