GeometricLocusMpolar::usage="Find the geometric locus of the point M of intersection of the polar of a fixed point P(γ, β)​, with respect to the circles that pass through two fixed points A(a, 0) and B(-a, 0) with the segment that joins P(γ, β) with the center of the circle. References: Carrillo Hoyo, Ángel et al. Lugares Geométricos. Instituto de Matemáticas, Facultad de Ciencias, UNAM; Personal.​
";
Development:
If we form a triangle with the upper intersection point of the polar with the circumference, the proportionality condition is met (r is the radius of circumference, the dstance of C to upper intersection point of the polar line with the circumference):


CP

r
=
r


CM

;
2
r
=

CP


CM
;
By the proportionality condition between vectors (for simplicity of the demonstration we are going to assume that C is at the origin):

CM
=k

CP
;​​k
2


CP

=
2
r
;k=
2
r
2


CP

;​​M={x',y'};P={x,y};​​x'=kx;y'=ky;​​x'=
2
r
2
x
+
2
y
x;y'=
2
r
2
x
+
2
y
y;
But C = {
x
0
,
y
0
}
it is not at the origin, so moving the axes:
M=
x
0
+
2
r
2
(x-
x
0
)
+
2
(y-
y
0
)
(x-
x
0
),
y
0
+
2
r
2
(x-
x
0
)
+
2
(y-
y
0
)
(y-
y
0
);
Note: For the inversion point M, the inversion coordinates were used and the values
x
0
= 0,
y
0
= t, a = 35, x = 30, y= 45
(the point P(x = γ, y =β)) were also used.
​
Animation:
In[]:=
ManipulateA={35,0};B={-35,0};C1={0,t1};P={30,45};r[t_]:=
2
35
+
2
t
;​​x1[t_]:=
2
r[t]
(30)
2
30
+
2
(45-t)
;​​y1[t_]:=t+
2
r[t]
(45-t)
2
30
+
2
(45-t)
;M[t_]:={x1[t],y1[t]};polar[x_]:=y1[t1]-
x1[t1]-30
y1[t1]-45
(x-x1[t1]);​​Show[{​​ParametricPlot[M[t],{t,-200,t1},ColorFunction->"BlueGreenYellow"],​​Plot[Callout[polar[x],"Polar of P",{Scaled[0.75],Below},LabelStyle{14,Bold,Black}],{x,-200,200},PlotStyle{Black,Dashed}],​​Graphics[{Black,PointSize[0.020],​​Point[A],Text[Style["A",14,Bold],A,{0,-1.2}],​​Point[B],Text[Style["B",14,Bold],B,{0,-1.2}],​​Point[C1],Text[Style["C",14,Bold],C1,{0,-1.2}],​​Point[P],Text[Style["P",14,Bold],P,{0,-1.2}],​​Point[M[t1]],Text[Style["M",14,Bold],M[t1],{0,-1.2}],Thick,Line[{P,C1}],Line[{C1,M[t1]}],Blue,Circle[C1,r[t1]]}]},AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange{{-200,200},{-400,400}},ImageSize300],Style["Geometric locus of M ",Bold,Large],{{t1,-200+0.00002,"Value (t)"},-200+0.000001,200,0.00001},ControlPlacementTop
Out[]=
​
Geometric locus of M
Value (t)