Perpendicular Tangents to a Parabola

pbtnlns.html

Exploration

Show that if and are two lines tangent to a parabola that intersect on the directrix of the parabola, then and are perpendicular to each other.

Approach

Since the shape (not the position or orientation) of the parabola is relevant, pick a parabola in standard position and a point on the parabola's directrix. Construct the tangent lines from the point to the parabola and show that the lines are perpendicular (i.e. their slopes are negative reciprocals).

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 and its directrix.

Clear[f];
parab1=Parabola2D[{0,0},f,0];
dln=First[Directrices2D[parab1]]

Construct a general point on the directrix.

Clear[y];
p1=Point2D[-f,y];

Construct the two tangent lines from the point.

{l1,l2}=TangentLines2D[p1,parab1] //Simplify

Show that the slopes are negative reciprocal (therefore the lines are perpendicular to each other).

Slope2D[l1]*Slope2D[l2] //Simplify

Discussion

This is the plot of a numerical example.

Sketch2D[{parab1,dln,p1,l1,l2} /.
         {f->1,y->2},
         CurveLength2D->20]

Graphics saved as "pbtnln01.eps".