Circle of Apollonius

apollon.html

Exploration

Show that the locus of a point P(x,y) that moves so that the ratio of its distance from two fixed points and is a circle with radius

and center

where . The locus is called the Circle of Apollonius for the points and and the ratio k.

Approach

Form the equation of the locus directly from the conditions. Show that the locus is the circle described.

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 points.

Clear[x1,y1,x2,y2,x,y];
P1=Point2D[{x1,y1}];
P2=Point2D[{x2,y2}];
P=Point2D[{x,y}];

Compute the distances.

d1=Distance2D[P1,P];
d2=Distance2D[P2,P];

Form the equation representing the relationship.

Clear[k]
eq1=k^2*d2^2-d1^2 //Expand

Construct the circle from its equation. The numerator under the radical is d k.

C1=Circle2D[Quadratic2D[eq1,{x,y}]] //FullSimplify

Clear[d,E1,E2,E3];
C2=C1 //. {
   k^2*((x1-x2)^2+(y1-y2)^2)-> d^2*k^2,
   Sqrt[E1_^2*E2_^2*E3_]->E1*E2/Sqrt[1/E3]}

Discussion

This is a plot of a numerical example with , and k=1.5.

d=Distance2D[P1,P2];
Sketch2D[{P1,P2,C2} //. {
   x1->1, y1->1, x2->-1, y2->-2, k->1.5}]

Graphics saved as "apollo01.eps".