Length of Hyperbola Focal Chord

hyplen.html

Exploration

Prove that the length of the focal chord of a hyperbola is , where a is the length of the semi-transverse axis and b is the length of the semi-conjugate axis.

Approach

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

Clear[a1,b1];
h1=Hyperbola2D[{0,0},a1,b1,0];

Construct one of the focal points.

fpt=First[Foci2D[h1]]

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

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

Intersect the line with the hyperbola.

pts=Points2D[fln,h1]

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

d1=Distance2D[Sequence @@ pts]

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

d1 /. {Sqrt[b1^4/a1^2]->b1^2/a1}