Circle Tangent to a Parabola

pbtancir.html

Exploration

Any line through the point (-3a,0) cuts the parabola in the points P and Q. Prove that the circle through P, Q and the focus is tangent to the parabola.

Approach

Construct the geometry and show that the lines tangent to the parabola and the circle at the intersection point are coincident.

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 point, parabola and a line through the point.

Clear[a,m];
p1=Point2D[-3a,0];
parab1=Parabola2D[{0,0},a,0];
l1=Line2D[p1,m]

Intersect the lines in pairs to find the intersection points, P and Q.

{P,Q}=Points2D[l1,parab1] //Simplify

Construct the circle through P, Q and the focus.

fpt=Foci2D[parab1][[1]];
c1=Circle2D[P,Q,fpt] //Simplify

Intersect the circle and the parabola.

pts=Points2D[c1,parab1] //Simplify

Two of the points are P and Q, as expected; the third must be the tangency point.

pts=Points2D[c1,parab1] //Simplify

Two of the points are P and Q, as expected; the third must be the tangency point.

{IsCoincident2D[pts[[2]],P], IsCoincident2D[pts[[3]],Q]}

Construct the tangents to the circle and the parabola at the intersection point.

{tln1=Line2D[pts[[1]],c1],
tln2=Line2D[pts[[1]],parab1]} //Simplify

The tangents are coincident, therefore, the parabola and the circle are tangent.

Map[SimplifyCoefficients2D[List @@ #]&,
    {tln1,tln2}]

Discussion

This is the plot of a numerical example.

Sketch2D[{c1,fpt,p1,parab1,l1,P,Q,pts[[1]]} /.
         {a->2/3,m->1/2},
   PlotRange->{{-3,10},{-7,7}},
   CurveLength2D->25]

Graphics saved as "pbtanc01.eps".