​
AQuadricGeometricLocus::usage="Find the equation of the geometric locus of the points of space such that the ratio of their distances to the plane x + y + z = 9 and to the point {1, 1, 1} is
3
. Check that it is a quadric, classify it and find its reduced equation. References: htpps://fernandorevilla.es; Personal.";
Development:
In[]:=
x+y+z-9
3
2
(x-1)
+
2
(y-1)
+
2
(z-1)
==
3
;​​Expand[
2
(x+y+z-9)
==9(
2
(x-1)
+
2
(y-1)
+
2
(z-1)
)]
81-18x+
2
x
-18y+2xy+
2
y
-18z+2xz+2yz+
2
z
27-18x+9
2
x
-18y+9
2
y
-18z+9
2
z
;​​8
2
x
+8
2
y
+8
2
z
-2xy-2xz-2yz-54==0;​​4
2
x
+4
2
y
+4
2
z
-xy-xz-yz-27==0;
In[]:=
Det[{{8,-1,-1,0},{-1,8,-1,0},{-1,-1,8,0},{0,0,0,-54}}]
Out[]=
-26244
In[]:=
Det[{{8,-1,-1},{-1,8,-1},{-1,-1,8}}]
Out[]=
486
In[]:=
A
A
44
==
-26244
486
A
A
44
-54;
In[]:=
Eigenvalues[{{8,-1,-1},{-1,8,-1},{-1,-1,8}}]
Out[]=
{9,9,6}
6
2
x'
+9
2
y'
+9
2
z'
==54;​​
2
x'
9
+
2
y'
6
+
2
z'
6
==1;
It’s an ellipsoid.
The domain of the ellipsoid is obtained:
In[]:=
Reduce[
2
(-x-y)
-16(4
2
x
+4
2
y
-xy-27)>=0,{x,y}]
Out[]=
x-
7
&&y-
1
7
||-
7
<x<
7
&&
x
7
-
4
7
3
7-
2
x
≤y≤
x
7
+
4
7
3
7-
2
x
||x
7
&&y
1
7
Orthogonal projection of the ellipsoid point on the plane:
In[]:=
f1[x_,y_]:=
x+y+
2
(-x-y)
-16(4
2
x
+4
2
y
-xy-27)
8
;​​Solve
x-x1
1
==
y-y1
1
==
z-f1[x1,y1]
1
,x+y+z-9==0,{x,y,z}
x
1
8
24+5x1-3y1-
48-7
2
x1
+2x1y1-7
2
y1
,y
1
8
24-3x1+5y1-
48-7
2
x1
+2x1y1-7
2
y1
,z
1
4
12-x1-y1+
48-7
2
x1
+2x1y1-7
2
y1
;
In[]:=
SwatchLegend[{Red,Black,Green},{"Point on the ellipsoid","Point on the plane","The point {1, 1, 1} is not on the ellipsoid"}]​​Manipulatef[x_,y_,z_]:=4
2
x
+4
2
y
+4
2
z
-xy-xz-yz-27;​​g[x_,y_,z_]:=x+y+z-9;f1[x_,y_]:=
x+y+
2
(-x-y)
-16(4
2
x
+4
2
y
-xy-27)
8
;g1[x_,y_]:=9-x-y;dp[x_,y_]:=Abs
x+y+f1[x,y]-9
3
;dp1[x_,y_]:=
2
(x-1)
+
2
(y-1)
+
2
(f1[x,y]-1)
;r[x_,y_]:=
dp[x,y]
dp1[x,y]
;​​y1=
x1
7
;​​​​w=Grid[{{PointCoordinates,PlaneDistance,PointDistance,Rate},{{x1,y1,f1[x1,y1]},dp[x1,y1],dp1[x1,y1],r[x1,y1]}},FrameAll,ItemSize->15,ItemStyle->15];ShowContourPlot3D[f[x,y,z]==0,{x,-5,5},{y,-5,5},{z,-5,5},PerformanceGoal"Quality",MeshStyleDirective[AbsoluteThickness[0.5],Opacity[0.2]],ContourStyleOpacity[0.1],PlotPoints->60],ContourPlot3D[g[x,y,z]==0,{x,-10,10},{y,-10,10},{z,-10,10},Mesh->None,ContourStyleDirective[Gray,Opacity[0.1],Specularity[White,30]],PerformanceGoal"Quality"],Graphics3DBlack,Ball
1
8
24+5x1-3y1-
48-7
2
x1
+2x1y1-7
2
y1
,
1
8
24-3x1+5y1-
48-7
2
x1
+2x1y1-7
2
y1
,
1
4
12-x1-y1+
48-7
2
x1
+2x1y1-7
2
y1
,0.17,Graphics3D[{Green,Ball[{1,1,1},0.17]}],Graphics3D[{Red,Ball[{x1,y1,f1[x1,y1]},0.17]}],Graphics3D[{Black,Dashed,Thick,Line[{{x1,y1,f1[x1,y1]},{1,1,1}}]}],Graphics3DBlack,Dashed,Thick,Line{x1,y1,f1[x1,y1]},
1
8
24+5x1-3y1-
48-7
2
x1
+2x1y1-7
2
y1
,
1
8
24-3x1+5y1-
48-7
2
x1
+2x1y1-7
2
y1
,
1
4
12-x1-y1+
48-7
2
x1
+2x1y1-7
2
y1
,AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize700,ViewPoint{4,-6,1},Style["Ellipsoid:",Bold,Large],x1,-
7
+0.02,"Value (x)",-
7
+0.01,
7
-0.01,0.00001,Delimiter,{{w,2,"Coordinates of the point P on the ellipsoid and ratio to its distances to the plane and point {1, 1, 1}"}},SaveDefinitions->True,ControlPlacementTop
Out[]=
Point on the ellipsoid
Point on the plane
The point {1, 1, 1} is not on the ellipsoid
Out[]=
​
Ellipsoid:
Value (x)
Coordinates of the point P on the ellipsoid and ratio to its distances to the plane and point {1, 1, 1}
PointCoordinates
PlaneDistance
PointDistance
Rate
{0.998219,0.142603,2.54867}
3.06603
1.77017
1.73205