Lösungen/ Solutions

Remove["Global`*"]

Aufgabe 1 /Problème 1

Remove["Global`*"]

3  x^2 + 3  y^2 - 30  x + 18  y + 54 == 0

?Collect

expr = 
      3  x^2 + 3  y^2 - 30  x + 18  y + 54 - 
        Expand[3 (x - k1)^2 + 3 (y - k2)^2 + r]

Collect[expr, {x, x^2, y, y^2}]

solv1 = Solve[-30 + 6 k1 == 0, {k1}] // Flatten

solv2 = Solve[18 + 6 k2 == 0, {k2}] // Flatten

expr1 = expr  /. solv1 /. solv2

solv3 = Solve[expr1 == 0, {r}] // Flatten

3 (x - k1)^2 + 3 (y - k2)^2 + r /. solv1 /. solv2 /. 
      solv3

{M = {5, -3},  r = Sqrt[48/3]}

Aufgabe 2 /Problème 2

Remove["Global`*"]

HNF1[x_, y_] := (1/(-6) x  +  1/5 y  - 1)* 
        1/Sqrt[(1/(-6))^2 +  ( 1/5)^2]

HNF2[x_, y_] := (1 x  +  1/(-2) y  - 1)* 
        1/Sqrt[1^2 +  ( 1/(-2))^2]

<< Graphics`ImplicitPlot`

Options[ImplicitPlot]

?PlotRange

ImplicitPlot[{HNF1[x, y] == 0, HNF2[x, y] == 0, 
          HNF1[x, y] == HNF2[x, y]}, {x, -7, 6}, 
        PlotRange -> {-3, 10}];

a

d0 =  HNF1[0, 0]

N[%]

b

n[t_] := t {-1/6, 1/5}

solv = 
      Solve[{HNF1[x, y] == 0, {x, y} == n[t]}, {x, y, t}] // 
        Flatten

N[%]

P0 = {x, y} /. solv

N[%]

c

HNF1[{x_, y_}] := HNF1[x, y]

U = {-2, 7};  
dU =  HNF1[U]

N[%]

d

solv3 = 
      Solve[HNF1[0, y] == HNF2[0, y], {y}] // Flatten

Q = {0, y /. solv3}

N[%]

e

solv4 = 
      Solve[{HNF1[x, y] == 0, HNF2[x, y] == 0}, {x, y}] // 
        Flatten

S = {x, y} /. solv4

f

vecLen[a_] := Sqrt[a . a]

n1 = {-1/6, 1/5};  n2 = {1, -1/2};

fi1 = ArcCos[-n1 . n2/(vecLen[n1] vecLen[n2])]

N[%]

%* 360/(2 Pi)

fi2 = ArcCos[+n1 . n2/(vecLen[n1] vecLen[n2])]

N[%]

%* 360/(2 Pi)

fi1 + fi2 // N

%* 360/(2 Pi)

Aufgabe 3 /Problème 3

a

n1 = {3, 2, -3};  n2 = {4, -2, 1};

fi = ArcCos[+n1 . n2/(vecLen[n1] vecLen[n2])]

N[%]

%* 360/(2 Pi)

b

solv = 
      Solve[{3  x + 2  y - 3*0 == 4, 
            4  x - 2  y + 0 == -1}, {x, y}] //  Flatten

P0 = {x, y} /.  solv

N[%]

Aufgabe 4 /Problème 4

a

S1 = ((x - 2)^2 + (y - 5)^2 == 4)

S2 = ((x - 5)^2 + (y - 4)^2 == 9)

solv = Solve[{S1, S2}, {x, y}]

N[%]

F1 = {x, y}  /.  solv[[1]]

N[%]

F2 = {x, y}  /.  solv[[2]]

N[%]

b

M1 = {2, 5}

Tg[x_, y_] :=  (F1 - M1) . ({x, y} - M1) - 4  //  
          Expand;
Collect[Tg[x, y], {x, y}]

Tg[x, y] // N

HNF[{x_, y_}] := 
      Tg[x, y]/vecLen[{((3/4) - ((3 (Sqrt[(3/5)]))/4)) , (-(1/4) - ((9 (Sqrt[(3/5)]))/4))}]

-HNF[F2]

N[%]