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Length of Ellipse Focal Chord

elllen.html

Exploration

Prove that the length of the focal chord of an ellipse is "elllen_1.gif", where a is the length of the semi-major axis and b is the length of the semi-minor axis.

Approach

Construct an ellipse in standard position. Construct a line perpendicular to the axis of the ellipse through one of the focal points (the line containing the focal chord). Compute the distance between the points of intersection of the ellipse and the line.

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.

Clear[a1,b1];
e1=Ellipse2D[{0,0},a1,b1,0];

Construct one of the focal points.

fpt=First[Foci2D[e1]]

"elllen_2.gif"

Construct a line perpendicular to the x-axis through the focus.

fln=Line2D[fpt,Infinity]

"elllen_3.gif"

Intersect the line with the ellipse.

pts=Points2D[fln,e1]

"elllen_4.gif"

The length of the focal chord is the distance between the intersection points.

d=Distance2D[Sequence @@ pts]

"elllen_5.gif"

Notice that since a>0 and b>0 the solution reduces to "elllen_6.gif".

d /. {Sqrt[b1^4/a1^2]->b1^2/a1}

"elllen_7.gif"


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