Length of Parabola Focal Chord

pbfocchd.html

Exploration

Prove that the length of the focal chord of a parabola is 4f, where f is the focal length.

Approach

Construct a parabola in a standard position. Construct a line perpendicular to the axis of the parabola through the focus point (the line containing the focal chord). Compute the distance between the points of intersection of the parabola 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 parabola.

Clear[f1];
par1=Parabola2D[{0,0},f1,0];

Construct the focus point.

fpt=First[Foci2D[par1]]

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

fln=Line2D[fpt,Line2D[0,1,0],Perpendicular2D]

Intersect the line with the parabola.

pts=Points2D[fln,par1]

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

Distance2D[Sequence @@ pts] /.
   Sqrt[f1^2]->f1