Line Normal to a Quadratic

lnquad.html

Exploration

Show that the normal line passing through the point on the quadratic whose equation is is given by

where

and .

Approach

Construct the polar line of the quadratic with respect to the quadratic. Construct the line normal to the polar through the point.  This is the desired normal 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

Construct the polar line (which is tangent to the quadratic if the point is on the quadratic).

Clear[x1,y1,a,b,c,d,e,f];
p1=Point2D[x1,y1];
q1=Quadratic2D[a,b,c,d,e,f];
l1=Line2D[p1,q1]

Construct the normal line.

l2=Line2D[p1,l1] //Simplify

Discussion

Define a function for constructing the normal line.

Line2D[
   p1:Point2D[{x1_,y1_}],
   q1:Quadratic2D[a_,b_,c_,d_,e_,f_],
   Normal2D] :=
Simplify[Line2D[p1,Line2D[p1,q1]]];

This is the plot of a numerical example.

q1=Quadratic2D[e1=Ellipse2D[{0,0},2,1,0]];
p1=Point2D[e1[Pi/9]];
l1=Line2D[p1,q1,Normal2D]

Sketch2D[{e1,p1,l1}, CurveLength2D->7]

Graphics saved as "lnquad01.eps".