Sum of Focal Distances of an Ellipse

ellips2a.html

Exploration

Show that the sum of the distances from the two foci to any point on an ellipse is 2a, where a is the length of the semi-major axis.

Approach

Construct a generic point on an ellipse.  Construct the two foci of the ellipse. Find the distance from each focus to the generic point.  Show that the sum of the distances is 2a.

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

Create the ellipse and a generic point on it.

Clear[a,b,t];
e1=Ellipse2D[{0,0},a,b,0];
p1=Point2D[e1[t]]

Construct the focus points of the ellipse.

{f1,f2}=Foci2D[e1]

Find the sum of the distances from the generic point to the foci.

sum1=Distance2D[p1,f1]+Distance2D[p1,f2]

Work on the expressions under the radicals.

{e1,e2}=Map[Expand[# /. Sin[t]^2->1-Cos[t]^2]&,
            {sum1[[1,1]],sum1[[2,1]]}]

This shows that both expressions factor into perfect squares.

{(e3=(a-Sqrt[a^2-b^2]*Cos[t])^2)-e1,
(e4=(a+Sqrt[a^2-b^2]*Cos[t])^2)-e2} //Expand

Replace the expressions under the radicals with the equivalent perfect square expressions.

sum2=Sqrt[e3]+Sqrt[e4]

Since a>b>0, both expressions are clearly the square root of a squared positive number, which simply reduces to the positive number itself.

Clear[E1];
sum2 /. Sqrt[E1_^2]->E1