Radical Axis Ratio

raratio.html

Exploration

Show that the point of intersection of the radical axis and the line of centers of two circles of radii and divides the segment between the two centers into the ratio

where d is the distance between the centers.

Approach

Create the two circles in a simplified, but sufficiently general, position.  Construct the radical axis and intersect it with the line segment between the centers.  Inspect the appropriate ratio.

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 two circles, one with center at the origin, the other with center at (d,0).

Clear[r1,r2,d];
c1=Circle2D[{0,0},r1];
c2=Circle2D[{d,0},r2];

Construct the radical axis of the two circles.

l1=Line2D[c1,c2]

Intersect the radical axis with the x-axis to find the point of division.

pt=Point2D[l1,Line2D[0,1,0]]

Form the desired ratio.

ratio1=Distance2D[Point2D[0,0],pt]/
       Distance2D[pt,Point2D[d,0]] //Simplify

Since all the expressions under the radical are positive, we can simplify the radicals.

Clear[E1,E2];
ratio2=ratio1 //.  {
   Sqrt[E1_^2/E2_^2]->E1/E2,
   1/Sqrt[E1_^2/E2_^2]->E2/E1}