Development:
2
x
+
2
y
-r=
2
(x-a)
+
2
y
;​​
2
x
+
2
y
=r+
2
(x-a)
+
2
y
;
Raising both members squared and developing remains:
2ax-
2
r
-
2
a
=2r
2
(x-a)
+
2
y
Let a = 6 and r = 2, then:
12x-40=4
2
(x-6)
+
2
y
;​​3x-10=
2
(x-6)
+
2
y
;
Raising both members squared and developing remains:
64-48x+8
2
x
-
2
y
=0;
In[]:=
Manipulater=2;P[t_]:=Cosh[t]+3,2
2
Sinh[t];​​P1[t_]:=2
3
9+6Cosh[t]+
2
Cosh[t]
+8
2
Sinh[t]
+
Cosh[t]
9+6Cosh[t]+
2
Cosh[t]
+8
2
Sinh[t]
,
4
2
Sinh[t]
9+6Cosh[t]+
2
Cosh[t]
+8
2
Sinh[t]
;​​​​O1={0,0};A1={6,0};d1=Norm[P[x1]-P1[x1]];d2=Norm[P[x1]-A1];​​t=Grid[{{
PP
1
,
PA
},{N[d1],N[d2]}},FrameAll,ItemStyle->15,ItemSize15];​​ShowContourPloty-
4
2
Sinh[x1]
9+6Cosh[x1]+
2
Cosh[x1]
+8
2
Sinh[x1]
==-
Cosh[x1]+3
2
2
Sinh[x1]
x-2
3
9+6Cosh[x1]+
2
Cosh[x1]
+8
2
Sinh[x1]
+
Cosh[x1]
9+6Cosh[x1]+
2
Cosh[x1]
+8
2
Sinh[x1]
,{x,-7,7},{y,-7,7},ContourStyle{Black,Dashed},​​ParametricPlotP[x],x,-
π
2
,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],Text[Style["A",14,Bold],A1,{0,-1.2}],Thick,Line[{P[x1],A1}],​​Line[{P[x1],P1[x1]}],Blue,Thick,Circle[O1,r]}],AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange7,Style["Right branch of a hyperbola",Bold,Large],x1,-
π
2
+0.00002,"Value (​
x
1
​)",-
π
2
+0.000001,
π
2
,0.00001,Delimiter,{{t,3,"Distance from point P to circumference (​
P
1
​) and from point P to point A "}},ControlPlacementTop
Out[]=
​
Right branch of a hyperbola
Value (​
x
1
​)
Distance from point P to circumference (​
P
1
​) and from point P to point A
PP
1
PA
6.52752
6.52752