Exploring Analyic Geometry with Mathematica®

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Circle Tangent to Circle, Center on Circle, Radius

tancir2.html

Exploration

Show that the centers (h,k) of the two circles passing through the point "tancir2_1.gif" with center on the circle "tancir2_2.gif" and radius r=1 are given by

"tancir2_3.gif"

where "tancir2_4.gif". This is a special case of TangentCircles2D[{obj},ln | cir,r], where the object is a point.

Approach

The radius is given, r=1, so the center point (h,k) needs to be found. The equation "tancir2_5.gif" is formed noting that the given point is on the circle. The equation "tancir2_6.gif" is formed noting that the center is on this circle. Solve two equations in two unknowns.

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

Solve the two equations.

Clear[h,k,x1,y1,d1];
ans1=Solve[{(x1-h)^2+(y1-k)^2==1, h^2+k^2==1},
           {h,k}] //. {
        x1^2+y1^2->d1^2} //FullSimplify

"tancir2_7.gif"

Simplify. Without loss of generality, assume all the point coordinates are positive.

Clear[E1];
ans2=ans1 //. {
   x1^2*y1+y1^3->y1*d1^2,
   x1^2+y1^2->d1^2,
   Sqrt[d1^2*E1_]->d1*Sqrt[E1],
   Sqrt[x1^2*E1_]->x1*Sqrt[E1]} //FullSimplify

"tancir2_8.gif"

ans3=ans2 //. {
   d1^3-d1*y1^2->d1*(d1^2-y1^2),
   d1^2-y1^2->x1^2}

"tancir2_9.gif"

ans4=Map[Apart,ans3,3]

"tancir2_10.gif"


Copyright © 1999-2007 Donald L. Vossler, Descarta2D Publishing
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