Lösungen/Solutions/F1S2Ue/Ex15

L=Lektion/Leçon-- Sprechstunde/heures de consultation
W=Work/Arbeit/Travail
A  =Anhang/  Annexe / Appendice

Problem / Problème

Remove["Global`*"]

mF[x_] := MatrixForm[x];
T[x_] := Transpose[x];
vL[x_] := Sqrt[x . x]

Kegelschnitt? / 
            Section de cône?
-444 x + 
        66 (x^2) + 308 y - 24 x y + 59 (y^2) == -973

qf[x_, y_] := -444 x + 66 (x^2) + 308 y - 24 x y + 
        59 (y^2)

qf[x, y] == -973

<< Graphics`ImplicitPlot`

ImplicitPlot[
qf[x, 
            y] == -973, 
{x, 2, 5}, 
        AspectRatio -> 1];

x1 = x + 3; y1 = y - 2;
qf[x1, y1]  //  Expand

Expand[qf[x1, y1]] ==  -973

Simplify[Expand[qf[x1, y1]] ==  -973]

A = {{66, -12}, {-12, 59}};  mF[A]

vec[u_, v_] := {{u}, {v}};  mF[vec[u, v]]

qf1[u_, v_] := Flatten[T[vec[u, v]] . A . vec[u, v] ][[1]] // 
          Expand;
qf1[u, v]

qf1[u, v] == 1

Eigenvalues[A]

d = {{50, 0}, {0, 75}};  d // mF

ev = Eigenvectors[A]

x1 = ev[[1]]/vL[ev[[1]]]

x2 = ev[[2]]/vL[ev[[2]]]

X = T[{x1, x2}];  X // mF

m = X . d . Inverse[X]; m // mF

m == A

T[vec[u, v]] . m . vec[u, v] // Expand // Flatten // 
      First

mF[vec[u, v]] == mF[X . {{u1}, {v1}}]

mF[{{u1}, {v1}}] == mF[Inverse[X] . vec[u, v]]

s1 = 
      Solve[{{u1}, {v1}} == Inverse[X] . vec[u, v], {u, v}] // 
        Flatten

u = u /. s1[[1]]

v = v /. s1[[2]]

T[vec[u, v]] . m . vec[u, v] // Expand // Flatten // 
      First

(T[vec[u, v]] . m . vec[u, v] // Expand // Flatten // 
          First ) ==  1

ImplicitPlot[(T[vec[u, v]] . m . vec[u, v] // Expand // 
                Flatten // First ) ==  1, {u1, -10, 10}, 
        AspectRatio -> Sqrt[50/75]];

Remove["Global`*"]

Aufbau von / Construction de 
 - 444 x + 
        66 (x^2) + 308 y - 24 x y + 59 (y^2) == -973

mF[x_] := MatrixForm[x]

vecLen[x_] := Sqrt[x . x]

x1 = x - 3; y1 = y + 2;

X = 
      Transpose[{{3, 4}/vecLen[{3, 4}], {-4, 3}/vecLen[{-4, 3}]}]; 
    X // mF

d = {{2, 0}, {0, 3}}; d // mF

m = X . d . Inverse[X]; m // mF

v[u1_, u2_] := {{u1}, {u2}}; v[u1, u2] // mF

qf[v_] := Expand[Flatten[Transpose[v] . m . v]][[1]];  
    qf[v[u1, u2]]

qf[v[x, y]]

qf[v[x1, y1]] == 1

Expand[(-((444 x)/25) + ((66 (x^2))/25) + ((308 y)/25) - ((24 x y)/25) + ((59 (y^2))/25)) 25] == 1 - ((974/25)) 25

qf[v[x1, y1]] == 1 // Simplify

<< Graphics`ImplicitPlot`

ImplicitPlot[
-444 x + 66 (x^2) + 308 y - 
            24 x y + 59 (y^2) == -973, 
{x, 2, 5}, 
        AspectRatio -> 1];

Ueb/Ex 2      W

M = {{m11, m12}, {m21, m22}};  M // mF

g1 = (M . {{-4}, {0}} == {{1}, {3}})

g2 = (M . {{0}, {-4}} == {{5}, {-1}})

solv = Solve[{g1, g2}, {m11, m12, m21, m22}] // Flatten

M = M /. solv;  M // mF

OP = {{-4}, {-4}};  OQ = M . OP;  OQ //  mF

OS = {{u1}, {u2}};
M . OS == OS

Solve[M . OS == OS, {u1, u2}]

OS = u2 {{-1}, {1}};  OS // mF

SQ = OQ - OS; SQ // mF

SP = OP - OS; SP // mF

SQ == λ SP

solv4 = 
      Solve[SQ == λ SP, {u2, λ}]  // 
        Flatten

OS = u2 {{-1}, {1}} /. solv4;  OS // mF

λ = λ  /. solv4

Kontrolle / Contrôle

Eigenvalues[M]

Eigenvectors[M]