SphereCenterIntersectionLines::usage="A sphere ξ it has a center at the intersection point of the straight lines:​
ℒ
1
= {2, 2, 4} + t{6, 6, 7}​, t ϵ ,
ℒ
2
= {20, -10, 5} + r{-2, 3, 1}​, r ϵ and it's tangent to the plane π: 4x - 2y + 3z = O.Find:​(a) The cartesian equation of sphere ξ.​(b) The center of circumference Γ, which it is the intersection of ξ with another plane π' parallel to π, such that the center of ξ it's to
25
29
units of π'.References: Internet: Repositorio.pucp.pe, 1.2 Superficies Notables en
3

​, Ejercicios propuestos, prob.- 4; Personal. ";
(a) Getting the equation of the sphere:
In[]:=
Solve[{2+6t==20-2r,2+6t==-10+3r},{t,r}]
Out[]=
{{t1,r6}}
C={8,8,11}
In[]:=
r=
Abs[4(8)-2(8)+3(11)]
2
4
+
2
2
+
2
3
Out[]=
49
29
2
(x-8)
+
2
(y-8)
+
2
(z-11)
=
2401
29
;
(b) The center of circumference Γ:
In[]:=
Solve
25
29
==
4(8)-2(8)+3(11)+d
2
4
+
2
2
+
2
3
,d
Out[]=
{{d-24}}
In[]:=
Solve-
25
29
==
4(8)-2(8)+3(11)+d
2
4
+
2
2
+
2
3
,d
Out[]=
{{d-74}}
There are two solutions, there are two planes π’ and therefore two centers Γ of circumferences:
4x-2y+3z-24=0;​​4x-2y+3z-74=0;​​
ℒ
3
={8,8,11}+t{4,-2,3};
In[]:=
Solve[4(8+4t)-2(8-2t)+3(11+3t)-24==0,t]
Out[]=
t-
25
29

In[]:=
C
Γ
1
=8+4-
25
29
,8-2-
25
29
,11+3-
25
29

Out[]=

132
29
,
282
29
,
244
29

In[]:=
Solve[4(8+4t)-2(8-2t)+3(11+3t)-74==0,t]
Out[]=
t
25
29

C
Γ
2
=8+4
25
29
,8-2
25
29
,11+3
25
29

Out[]=

332
29
,
182
29
,
394
29

Parametric equations of the circumferences:
Circumference 1:
x=
1
2
y-
3
4
z+6;​​y=u;​​z=v;​​
1
2
,1,0;
In[]:=

1
2
,1,0{4,-2,3}
Out[]=
3,-
3
2
,-5
In[]:=
Solve
2
(x-8)
+
2
(y-8)
+
2
(z-11)
==
2401
29
,4x-2y+3z-24==0,{x,y}
Out[]=
x
1
290
2784-174z-
29
-24016+14152z-841
2
z
,y
1
290
2088+87z-2
29
-24016+14152z-841
2
z
,x
1
290
2784-174z+
29
-24016+14152z-841
2
z
,y
1
290
2088+87z+2
29
-24016+14152z-841
2
z

In[]:=
ξ[z_]:=
1
290
2784-174z-
29
-24016+14152z-841
2
z
,
1
290
2088+87z-2
29
-24016+14152z-841
2
z
,z;​​Reduce[-24016+14152z-841
2
z
>=0,z]
Out[]=
1
29
(244-8
555
)≤z≤
1
29
(244+8
555
)
In[]:=
r=Norm
132
29
,
282
29
,
244
29
-ξ[2]
Out[]=
34596
841
+
2
-
132
29
+
1
290
(2436-2
6699
)
+
2
282
29
+
1
290
(-2262+4
6699
)
ζ[t_]:=
132
29
,
282
29
,
244
29
+Cos[t]
34596
841
+
2
-
132
29
+
1
290
2436-2
6699

+
2
282
29
+
1
290
-2262+4
6699

Norm
1
2
,1,0

1
2
,1,0+Sin[t]
34596
841
+
2
-
132
29
+
1
290
2436-2
6699

+
2
282
29
+
1
290
-2262+4
6699

Norm3,-
3
2
,-5
3,-
3
2
,-5;
Circumference 2:
x=
1
2
y-
3
4
z+
37
2
;​​y=u;​​z=v;​​
1
2
,1,0;
In[]:=

1
2
,1,0{4,-2,3}
Out[]=
3,-
3
2
,-5
In[]:=
Solve
2
(x-8)
+
2
(y-8)
+
2
(z-11)
==
2401
29
,4x-2y+3z-74==0,{x,y}
Out[]=
x
1
290
5684-174z-
29
-119716+22852z-841
2
z
,y
1
290
638+87z-2
29
-119716+22852z-841
2
z
,x
1
290
5684-174z+
29
-119716+22852z-841
2
z
,y
1
290
638+87z+2
29
-119716+22852z-841
2
z

In[]:=
ξ
1
[z_]:=
1
290
5684-174z-
29
-119716+22852z-841
2
z
,
1
290
638+87z-2
29
-119716+22852z-841
2
z
,z;​​Reduce[-119716+22852z-841
2
z
>=0,z]​​
Out[]=
1
29
(394-8
555
)≤z≤
1
29
(394+8
555
)
In[]:=
r1=Norm
332
29
,
182
29
,
394
29
-
ξ
1
[8]
Out[]=
26244
841
+
2
-
332
29
+
1
290
(4292-2
67251
)
+
2
182
29
+
1
290
(-1334+4
67251
)
ζ
1
[t_]:=
332
29
,
182
29
,
394
29
+Cos[t]
26244
841
+
2
-
332
29
+
1
290
4292-2
67251

+
2
182
29
+
1
290
-1334+4
67251

Norm
1
2
,1,0

1
2
,1,0+Sin[t]
26244
841
+
2
-
332
29
+
1
290
4292-2
67251

+
2
182
29
+
1
290
-1334+4
67251

Norm3,-
3
2
,-5
3,-
3
2
,-5;
Graph (a):
Graph (b):