In[]:=
GeometricLocusSymbolicProblem::usage="D ϵ Ρ,
AD
⋂  = {E, B}. Find the geometric locus of G = MID(D, E) andH = MID(D, B)​, knowing that :
2
x
+
2
y
= 4, Ρ: y =
2
x
+1
2
​, focus: A{0, 1}​, directrix: y = 0.References: Internet, Facebook: A Beautiful Locus Problem with a Circle and a Parabola; Personal.";
Development:
Point D:
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t,
2
t
+1
2
;
Direction vector of
AD
:
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D-A==t,
2
t
+1
2
-1==t,
2
t
-1
2
;
Any point on the
AD
line:
ℒ: {0, 1} + λ
t,
2
t
-1
2

;
Obtaining the points E and B which are the intersection with the circle of the line
AD
:
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λt,1+λ
2
t
-1
2
;​​
2
(λt)
+
2
1+λ
2
t
-1
2
==4;​​Solve
2
(λt)
+
2
1+λ
2
t
-1
2
==4,λ;
In[]:=
λ
21-
2
t
-2
1+
2
t
+
4
t

1+2
2
t
+
4
t
,λ
21-
2
t
+2
1+
2
t
+
4
t

1+2
2
t
+
4
t
;
In[]:=
D1[t_]:=t,
2
t
+1
2
;​​B[t_]:=
21-
2
t
+2
1+
2
t
+
4
t

1+2
2
t
+
4
t
t,1+
21-
2
t
+2
1+
2
t
+
4
t

1+2
2
t
+
4
t
2
t
-1
2
;​​E1[t_]:=
21-
2
t
-2
1+
2
t
+
4
t

1+2
2
t
+
4
t
t,1+
21-
2
t
-2
1+
2
t
+
4
t

1+2
2
t
+
4
t
2
t
-1
2
;​​
D1[t]+E1[t]
2
//FullSimplify;​​
D1[t]+B[t]
2
//FullSimplify;
In[]:=

t3+
4
t
-4
1+
2
t
+
4
t

2
2
(1+
2
t
)
,
1
4
3+
2
t
-
2(-1+
2
t
)-1+
2
t
+2
1+
2
t
+
4
t

2
(1+
2
t
)
;
In[]:=

t3+
4
t
+4
1+
2
t
+
4
t

2
2
(1+
2
t
)
,
1
4
3+
2
t
+
2(-1+
2
t
)1-
2
t
+2
1+
2
t
+
4
t

2
(1+
2
t
)
;
Animation:
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ManipulateA={0,1};D1[t_]:=t,
2
t
+1
2
;B[t_]:=
21-
2
t
+2
1+
2
t
+
4
t

1+2
2
t
+
4
t
t,1+
21-
2
t
+2
1+
2
t
+
4
t

1+2
2
t
+
4
t
2
t
-1
2
;​​E1[t_]:=
21-
2
t
-2
1+
2
t
+
4
t

1+2
2
t
+
4
t
t,1+
21-
2
t
-2
1+
2
t
+
4
t

1+2
2
t
+
4
t
2
t
-1
2
;​​G[t_]:=
t3+
4
t
-4
1+
2
t
+
4
t

2
2
(1+
2
t
)
,
1
4
3+
2
t
-
2(-1+
2
t
)-1+
2
t
+2
1+
2
t
+
4
t

2
(1+
2
t
)
;​​H[t_]:=
t3+
4
t
+4
1+
2
t
+
4
t

2
2
(1+
2
t
)
,
1
4
3+
2
t
+
2(-1+
2
t
)1-
2
t
+2
1+
2
t
+
4
t

2
(1+
2
t
)
;​​Show​​ParametricPlot[{2Cos[t],2Sin[t]},{t,0,2π},ColorFunction->"DeepSeaColors"],​​ParametricPlott,
2
t
+1
2
,{t,-3,3},ColorFunction->"BrassTones",​​ParametricPlot[G[t],{t,-3,t1},ColorFunction->"AvocadoColors"],​​ParametricPlot[H[t],{t,-3,t1},ColorFunction->"BrownCyanTones"],​​Graphics[{Black,PointSize[0.015],​​Point[A],Text[Style["A",14,Bold],A,{0,-1.2}],​​Point[D1[t1]],Text[Style["D",14,Bold],D1[t1],{0,-1.2}],​​Point[B[t1]],Text[Style["B",14,Bold],B[t1],{0,-1.2}],​​Point[E1[t1]],Text[Style["E",14,Bold],E1[t1],{0,-1.2}],​​{Red,Point[G[t1]],Text[Style["G",14,Bold],G[t1],{0,-1.2}],​​Point[H[t1]],Text[Style["H",14,Bold],H[t1],{0,-1.2}]},​​{Orange,Thick,InfiniteLine[{E1[t1],B[t1]}]}}],AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange{{-3,3},{-3,6}},ImageSize500,Style["Geometric Locus of G and H",Bold,Large],{{t1,-3+0.00002,"Value (t)"},-3+0.000001,3,0.00001},AutorunSequencing->All,ControlPlacementTop
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​
Geometric Locus of G and H
Value (t)