Development:
2
(x-a)
+
2
y
=r-
2
x
+
2
y
;
Raising both members squared and developing remains:
-2ax-
2
r
+
2
a
=-2r
2
x
+
2
y
Be a = 3 and r = 5, then:
-6x-16=-10
2
x
+
2
y
;​​3x+8=5
2
x
+
2
y
;
Raising both members squared and developing remains:
-64-48x+16
2
x
+25
2
y
=0;
The point P is on the line
OC
and it must be fullfilled that
PC
=
PA
, then:
A={3,0};​​r=5;​​C1[θ_]:=r{Cos[θ],Sin[θ]};P[θ_,λ_]:=λC1[θ];​​Solve[Norm[P[θ,λ]-A]==Norm[P[θ,λ]-C1[θ]],λ]
In[]:=
λ
-9+25
2
Cos[θ]
+25
2
Sin[θ]
10(-3Cos[θ]+5
2
Cos[θ]
+5
2
Sin[θ]
)
;​​λ=
8
5(5-3Cos[θ])
;​​P[θ]=λC1[θ]

8Cos[θ]
5-3Cos[θ]
,
8Sin[θ]
5-3Cos[θ]
;
Manipulater=5;O1={0,0};A={3,0};​​P[θ_]:=
8Cos[θ]
5-3Cos[θ]
,
8Sin[θ]
5-3Cos[θ]
;​​C1[θ_]:=r{Cos[θ],Sin[θ]};d1=Norm[P[θ1]-C1[θ1]];d2=Norm[P[θ1]-A];​​t=Grid[{{
PC
,
PA
},{N[d1],N[d2]}},FrameAll,ItemStyle->15,ItemStyle->15];​​Show[{ContourPlot[y-5Sin[θ1]==-Cot[θ1](x-5Cos[θ1]),{x,-10,10},{y,-10,10},ContourStyle{Black,Dashed}],​​ParametricPlot[P[θ],{θ,0,θ1},ColorFunction->"BlueGreenYellow"],​​Graphics[{Black,PointSize[0.018],​​Point[O1],Text[Style["O",14,Bold],O1,{0,-1.2}],​​Point[P[θ1]],Text[Style["P",14,Bold],P[θ1],{0,-1.2}],​​Point[C1[θ1]],Text[Style["C",14,Bold],C1[θ1],{0,-1.2}],​​Point[A],Text[Style["A",14,Bold],A,{0,-1.2}],Thick,Line[{P[θ1],A}],​​Line[{O1,C1[θ1]}],Blue,Thick,Circle[O1,r]}]},AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},PlotRange10],Style["Ellipse",Bold,Large],{{θ1,0.00002,"Value (​
θ
1
​)"},0.000001,2π,0.1},Delimiter,{{t,3,"Distance from point P to circumference and from point P to point A. "}},ControlPlacementTop
Out[]=
​
Ellipse
Value (​
θ
1
​)
Distance from point P to circumference and from point P to point A.
PC
PA
2.9555
2.9555