Focus of Ellipse is Pole of Directrix

elfocdir.html

Exploration

Show that the focus of an ellipse is the pole of the corresponding directrix.

Approach

Construct the directrix and the pole of the focus and verify that they are the same lines.

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

Construct the required geometry.

Clear[a,b];
e1=Ellipse2D[{0,0},a,b,0];
fpts=Foci2D[e1];
dlns=Directrices2D[e1] //Simplify

Construct the polars of the foci.

lns={Line2D[fpts[[1]],e1], Line2D[fpts[[2]],e1]} //Simplify

The lines in pairs are coincident.

{IsCoincident2D[dlns[[1]],lns[[1]]],
IsCoincident2D[dlns[[2]],lns[[2]]]}

Discussion

This is a plot of a numerical example with a=1.5 and b=1.

Sketch2D[{e1,fpts,dlns} /. {a->1.5,b->1},
   CurveLength2D->3]

Graphics saved as "elfocd01.eps".