LugarGeometrico16Animate::usage="Find and identify the equation of the geometric locus of the center of a circle that is always tangent to the line y - 1 = 0 and to the circumference
2
x
+
2
y
= 9 (Two solutions). It's about the parabolas
2
x
- 4y - 4 = 0 y
2
x
+ 8y - 16 = 0. The animation shows how they are formed. References: Lehmann, Charles H. Geometría Analítica. Ed. Limusa: p.160, 2003;​ Personal. ";
Development:
2
x
+
2
y
-3==
|y-1|
1
;​​
2
x
+
2
y
-3==±(y-1)​​
Case 1:
In[]:=
2
x
+
2
y
-3==y-1;​​Expand
2

2
x
+
2
y

==
2
(y+2)

2
x
+
2
y
4+4y+
2
y
;​​
2
x
-4y-4==0;
Case 2:
2
x
+
2
y
-3==-(y-1);​​Expand
2

2
x
+
2
y

==
2
(-y+4)

2
x
+
2
y
16-8y+
2
y
;​​
2
x
+8y-16==0;
Animation:
In[]:=
Animateg1=Circlet,-
1
8
2
t
+2,Abs-
1
8
2
t
+1/.tt1;​​g6=Circlet,
1
4
2
t
-1,Abs-
1
4
2
t
+2/.tt1;​​g2=Pointt,
1
4
2
t
-1/.tt1;​​g3=Pointt,-
1
8
2
t
+2/.tt1;​​g7=Circle[{0,0},3];​​g4=ContourPlot[y-1==0,{x,-10,10},{y,-10,10}];​​g5=ParametricPlott,-
1
8
2
t
+2,{t,-6,t1},PlotStyleDashed,ColorFunction->"Aquamarine";​​g8=ParametricPlott,
1
4
2
t
-1,{t,-6,t1},PlotStyleDashed,ColorFunction->"Aquamarine";​​​​Show[{g4,g5,g8,Graphics[{Red,Thick,PointSize[0.018],g2}],Graphics[{Purple,Thick,PointSize[0.018],g3}],Graphics[{Black,g7}],Graphics[{Purple,Thick,g1}],Graphics[{Red,Thick,g6}]},AxesTrue,AxesOrigin{0,0},BoxRatiosAutomatic,PerformanceGoal"Quality"],{t1,-5.99,6,0.1},AnimationRunningFalse
Out[]=
t1