Confocal Parabolas

Author

Eric W. Weisstein
March 16, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/ConfocalParabolas.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ConfocalParabolas.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Figure

c=a e, so varying e and keeping c fixed means a=c/e.

Show[Graphics[{
Table[{PolarPlot[1/e(e^2-1)/(1-e Cos[θ]),{θ,-ArcTan[Sqrt[e^2-1]],ArcTan[Sqrt[e^2-1]]},Ticks->None,DisplayFunction->Identity][[1]],
PolarPlot[1/e(e^2-1)/(1-e Cos[θ]),{θ,ArcTan[Sqrt[e^2-1]],2Pi-ArcTan[Sqrt[e^2-1]]},Ticks->None,DisplayFunction->Identity][[1]]},
{e,1.02,2,.1}],
Line[{{-4,0},{4,0}}],
Line[{{-1,-2},{-1,2}}],
{PointSize[.03],Point/@{{0,0},{-2,0}}}
}],
DisplayFunction->$DisplayFunction,AspectRatio->Automatic,
PlotRange->{{-4,2},{-2,2}}]

b=.2;
a=.8;
c=Sqrt[a^2+b^2];
Table[{Sqrt[#.#]&[{a Sqrt[1+y^2/b^2]+c,y}],a^2/c+a Sqrt[1+y^2/b^2]},{y,-b+.1,b-.1,.05}]

Fit[%,{1,t},t]

{a,b,c}=.

Simplify[(a^2/c+a)/(a+c)]