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Apoapsis and Periapsis of an Ellipse



Show that the greatest, apoapsis, and least, periapsis, radial distance of a point on an ellipse as measured from a focus point is given by r=a(1+e) and r=a(1-e), respectively, where e is the eccentricity and a is the length of the semi-major axis of the ellipse.


Create a standard ellipse centered at the origin and create an expression representing the distance from a focus point to a point on the ellipse (in terms of the eccentricity and semi-major axis). Find the parameter value on the ellipse where the distance is a minimum or a maximum.


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The eccentricity is given by "ellrad_1.gif"; therefore, "ellrad_2.gif". Find the focus points and use this substitution.

fpts1=Foci2D[e1] /. Sqrt[a^2-b^2]->a*e


Find a general point on the ellipse in terms parameter t.



Solve for b in terms of a and e, where b>0.



Determine the distance, d, from the point to the focus.

d=Distance2D[fpts1[[1]],pt] /. ans[[2]] //Simplify


The maximum value of d occurs when cos (θ)=-1, or θ=π; the minimum value of d occurs when cos (θ)=1, or θ=0.

{apoapsis,periapsis}=Map[(d /. #)&,{Cos[t]->-1,Cos[t]->1}]


Since e<1, the sign must be reversed outside the radical.

{apoapsis,periapsis} /.
   Sqrt[a^2*E1_^2]->a*(-E1) //Factor


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