Line Tangent to a Circle

lntancir.html

Exploration

Show that the line is tangent to the circle for all values of m.

Approach

Show that the pole point (which is the point of tangency if the line is tangent to the circle) is on the circle.

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 line.

Clear[x,y,a,m];
l1=Line2D[y==m(x-a)+a*Sqrt[1+m^2],{x,y}]

Construct the circle.

c1=Circle2D[q1=Quadratic2D[x^2+y^2==2a*x,{x,y}]]

Construct the pole point.

p1=Point2D[l1,c1] //Simplify

The coordinates of the pole point satisfy the equation of the circle.

Polynomial2D[q1,Coordinates2D[p1]] //Simplify

Discussion

This is a plot of a numerical example.

Sketch2D[{c1 /. a->1,
   Map[({l1,p1} /. {a->1,m->#})&,
       {0,.5,1,2,5,-5,-2,-1,-.5}]},
   CurveLength2D->4]

Graphics saved as "lntanc01.eps".