Development:
Let P be of coordinates (x, y) and be
P
1
the point of contact between the normal and the tangent to the circumference then according to the figure and by the Pythagorean theorem, it is fulfilled that:
2
x
+
2
y
=
2


PP
1
+r
Then,based on the geometric locus described we know that

PP
1
= y, then:
2
x
+
2
y
=
2
(y+r)
Solving for y , the answer remains:
±y=
2
x
+
2
y
-r
Doing operations there are 2 equations left:
2
x
=2ry+
r
2
;
2
x
=-2ry-
r
2
;
In[]:=
Manipulater=4;P1[x_]:=
32x1
256+32
2
x1
+
4
x1
,4
2
x1
-16
16+
2
x1
;​​P[x_]:=x,
-
2
r
+
2
x
2r
;O1={0,0};A1[x_]:={x,0};d1=Norm[P[x1]-P1[x1]];d2=Norm[P[x1]-A1[x1]];​​t=Grid[{{
PP
1
,
PA
},{N[d1],N[d2]}},FrameAll,ItemStyle->15,ItemSize15];​​ShowContourPloty-4
2
x1
-16
16+
2
x1
==
2rx1
2
r
-
2
x1
x-
32x1
256+32
2
x1
+
4
x1
,{x,-7,7},{y,-7,7},ContourStyle{Black,Dashed},​​ParametricPlot[P[x],{x,-2π,x1},ColorFunction->"BlueGreenYellow"],​​Graphics[{Black,PointSize[0.018],​​Point[O1],Text[Style["O",14,Bold],O1,{0,-1.2}],​​Point[P[x1]],Text[Style["P",14,Bold],P[x1],{0,-1.2}],​​Point[P1[x1]],Text[Style["
P
1
",14,Bold],P1[x1],{0,-1.2}],​​Point[A1[x1]],Text[Style["A",14,Bold],A1[x1],{0,-1.2}],Thick,Line[{P[x1],A1[x1]}],​​Line[{P[x1],P1[x1]}],Blue,Thick,Circle[O1,r]}],AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange7,Style["Case 1 parabola",Bold,Large],{{x1,-2π+0.00002,"Value (​
x
1
​)"},-2π+0.000001,2π,0.00001},Delimiter,{{t,3,"Distance from point P to circumference (​
P
1
​) and from point P to point A "}},ControlPlacementTop
Out[]=
​
Case 1 parabola
Value (​
x
1
​)
Distance from point P to circumference (​
P
1
​) and from point P to point A
PP
1
PA
2.9348
2.9348
In[]:=
Manipulater=4;P1[x_]:=
32x1
256+32
2
x1
+
4
x1
,4
16-
2
x1
16+
2
x1
;​​P[x_]:=x,
2
r
-
2
x
2r
;O1={0,0};A1[x_]:={x,0};d1=Norm[P[x1]-P1[x1]];d2=Norm[P[x1]-A1[x1]];​​t=Grid[{{
PP
1
,
PA
},{N[d1],N[d2]}},FrameAll,ItemStyle->15,ItemSize15];​​ShowContourPloty-4
16-
2
x1
16+
2
x1
==
2rx1
2
x1
-
2
r
x-
32x1
256+32
2
x1
+
4
x1
,{x,-7,7},{y,-7,7},ContourStyle{Black,Dashed},​​ParametricPlot[P[x],{x,-2π,x1},ColorFunction->"BlueGreenYellow"],​​Graphics[{Black,PointSize[0.018],​​Point[O1],Text[Style["O",14,Bold],O1,{0,-1.2}],​​Point[P[x1]],Text[Style["P",14,Bold],P[x1],{0,-1.2}],​​Point[P1[x1]],Text[Style["
P
1
",14,Bold],P1[x1],{0,-1.2}],​​Point[A1[x1]],Text[Style["A",14,Bold],A1[x1],{0,-1.2}],Thick,Line[{P[x1],A1[x1]}],​​Line[{P[x1],P1[x1]}],Blue,Thick,Circle[O1,r]}],AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange7,Style["Case 2 parabola",Bold,Large],{{x1,-2π+0.00002,"Value (​
x
1
​)"},-2π+0.000001,2π,0.00001},Delimiter,{{t,3,"Distance from point P to circumference (​
P
1
​) and from point P to point A "}},ControlPlacementTop
Out[]=
​
Case 2 parabola
Value (​
x
1
​)
Distance from point P to circumference (​
P
1
​) and from point P to point A
PP
1
PA
2.9348
2.9348