GeometricLocusEllipses::usage="Find and identify the equation of the geometric locus of the center of a circle that remains tangent to the circles
2
x
+
2
y
- 4y - 12 = 0 and
2
x
+
2
y
= 1 (Two solutions). These are two ellipses: 100
2
x
+ 84
2
y
- 168 y - 441 = 0 and 36
2
x
+ 20
2
y
- 40 y - 25 = 0. The animation explains how they are formed. References: Lehmann, Charles H. Geometría Analítica. Ed. Limusa: p.186, 2003; Personal. ";
Development:
Case 1:
In[]:=
2
x
+
2
y
-1==4-
2
x
+
2
(y-2)
;​​Expand
2

2
x
+
2
y

==
2
5-
2
x
+
2
(y-2)

//FullSimplify
In[]:=
10
2
x
+
2
(-2+y)
+4y29;​​Expand
2
10
2
x
+
2
(-2+y)

==
2
(29-4y)

400+100
2
x
-400y+100
2
y
841-232y+16
2
y
;​​100
2
x
+84
2
y
-168y-441==0;
Case 2:
In[]:=
2
x
+
2
y
+1==4-
2
x
+
2
(y-2)
;​​Expand
2

2
x
+
2
y

==
2
3-
2
x
+
2
(y-2)

//FullSimplify
In[]:=
6
2
x
+
2
(-2+y)
+4y13;​​Expand
2
6
2
x
+
2
(-2+y)

==
2
(13-4y)

144+36
2
x
-144y+36
2
y
169-104y+16
2
y
;​​36
2
x
+20
2
y
-40y-25==0;
Animation:
In[]:=
Animateg1=Point
21
2
Sin[θ],
5
2
Cos[θ]+1/.θθ1;​​g2=Point
5
2
Sin[θ],
3
2
Cos[θ]+1/.θθ1;​​g3=Circle[{0,2},4];​​g4=Circle[{0,0},1];​​g5=ParametricPlot
21
2
Sin[θ],
5
2
Cos[θ]+1,
5
2
Sin[θ],
3
2
Cos[θ]+1,{θ,0,θ1},PlotStyleDashed,ColorFunction->"Aquamarine";​​xr=
21
2
Sin[θ1];yr=
5
2
Cos[θ1]+1;rr=
2
xr
+
2
yr
-1;​​xp=
5
2
Sin[θ1];yp=
3
2
Cos[θ1]+1;rp=
2
xp
+
2
yp
+1;​​g6=Circle[{xr,yr},rr];​​g7=Circle[{xp,yp},rp];​​​​Show[{g5,Graphics[{Red,Thick,PointSize[0.018],g1}],Graphics[{Purple,Thick,PointSize[0.018],g2}],Graphics[{Black,g3}],Graphics[{Black,g4}],Graphics[{Red,g6}],Graphics[{Purple,g7}]},AxesTrue,AxesOrigin{0,0},BoxRatiosAutomatic,PlotRangeAll,PerformanceGoal"Quality"],{θ1,0.01,2π,0.1},AnimationRunningFalse
Out[]=
θ1
In[]:=
​