Sr Examen
4x^2+y^2+2z^2+4xy+2sqrt(5)x-4sqrt(5)y-4z-18=0 forma canónica
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Solución
Ha introducido
[src]
2 2 2 ___ ___ -18 + y - 4*z + 2*z + 4*x - 4*y*\/ 5 + 2*x*\/ 5 + 4*x*y = 0
$$4 x^{2} + 4 x y + 2 \sqrt{5} x + y^{2} - 4 \sqrt{5} y + 2 z^{2} - 4 z - 18 = 0$$
4*x^2 + 4*x*y + 2*sqrt(5)*x + y^2 - 4*sqrt(5)*y + 2*z^2 - 4*z - 18 = 0
Método de invariantes
Se da la ecuación de superficie de 2 grado:
$$4 x^{2} + 4 x y + 2 \sqrt{5} x + y^{2} - 4 \sqrt{5} y + 2 z^{2} - 4 z - 18 = 0$$
Esta ecuación tiene la forma:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0$$
donde
$$a_{11} = 4$$
$$a_{12} = 2$$
$$a_{13} = 0$$
$$a_{14} = \sqrt{5}$$
$$a_{22} = 1$$
$$a_{23} = 0$$
$$a_{24} = - 2 \sqrt{5}$$
$$a_{33} = 2$$
$$a_{34} = -2$$
$$a_{44} = -18$$
Las invariantes de esta ecuación al transformar las coordenadas son los determinantes:
$$I_{1} = a_{11} + a_{22} + a_{33}$$
$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|$$
sustituimos coeficientes
$$I_{1} = 7$$
$$I_{3} = \left|\begin{matrix}4 & 2 & 0\\2 & 1 & 0\\0 & 0 & 2\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}4 & 2 & 0 & \sqrt{5}\\2 & 1 & 0 & - 2 \sqrt{5}\\0 & 0 & 2 & -2\\\sqrt{5} & - 2 \sqrt{5} & -2 & -18\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}4 - \lambda & 2 & 0\\2 & 1 - \lambda & 0\\0 & 0 & 2 - \lambda\end{matrix}\right|$$
$$I_{1} = 7$$
$$I_{2} = 10$$
$$I_{3} = 0$$
$$I_{4} = -250$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 7 \lambda^{2} - 10 \lambda$$
$$K_{2} = -155$$
$$K_{3} = -375$$
Como
$$I_{3} = 0 \wedge I_{2} \neq 0 \wedge I_{4} \neq 0$$
entonces por razón de tipos de rectas:
hay que
Formulamos la ecuación característica para nuestra superficie:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
o
$$\lambda^{3} - 7 \lambda^{2} + 10 \lambda = 0$$
$$\lambda_{1} = 5$$
$$\lambda_{2} = 2$$
$$\lambda_{3} = 0$$
entonces la forma canónica de la ecuación será
$$\tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
y
$$- \tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
$$5 \tilde x^{2} + 2 \tilde y^{2} + 10 \tilde z = 0$$
y
$$5 \tilde x^{2} + 2 \tilde y^{2} - 10 \tilde z = 0$$
$$2 \tilde z + \left(\frac{\tilde x^{2}}{1} + \frac{\tilde y^{2}}{\frac{5}{2}}\right) = 0$$
y
$$- 2 \tilde z + \left(\frac{\tilde x^{2}}{1} + \frac{\tilde y^{2}}{\frac{5}{2}}\right) = 0$$
es la ecuación para el tipo paraboloide elíptico
- está reducida a la forma canónica
$$4 x^{2} + 4 x y + 2 \sqrt{5} x + y^{2} - 4 \sqrt{5} y + 2 z^{2} - 4 z - 18 = 0$$
Esta ecuación tiene la forma:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0$$
donde
$$a_{11} = 4$$
$$a_{12} = 2$$
$$a_{13} = 0$$
$$a_{14} = \sqrt{5}$$
$$a_{22} = 1$$
$$a_{23} = 0$$
$$a_{24} = - 2 \sqrt{5}$$
$$a_{33} = 2$$
$$a_{34} = -2$$
$$a_{44} = -18$$
Las invariantes de esta ecuación al transformar las coordenadas son los determinantes:
$$I_{1} = a_{11} + a_{22} + a_{33}$$
|a11 a12| |a22 a23| |a11 a13|
I2 = | | + | | + | |
|a12 a22| |a23 a33| |a13 a33|$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|$$
|a11 a14| |a22 a24| |a33 a34|
K2 = | | + | | + | |
|a14 a44| |a24 a44| |a34 a44| |a11 a12 a14| |a22 a23 a24| |a11 a13 a14|
| | | | | |
K3 = |a12 a22 a24| + |a23 a33 a34| + |a13 a33 a34|
| | | | | |
|a14 a24 a44| |a24 a34 a44| |a14 a34 a44|sustituimos coeficientes
$$I_{1} = 7$$
|4 2| |1 0| |4 0|
I2 = | | + | | + | |
|2 1| |0 2| |0 2|$$I_{3} = \left|\begin{matrix}4 & 2 & 0\\2 & 1 & 0\\0 & 0 & 2\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}4 & 2 & 0 & \sqrt{5}\\2 & 1 & 0 & - 2 \sqrt{5}\\0 & 0 & 2 & -2\\\sqrt{5} & - 2 \sqrt{5} & -2 & -18\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}4 - \lambda & 2 & 0\\2 & 1 - \lambda & 0\\0 & 0 & 2 - \lambda\end{matrix}\right|$$
| ___| | ___|
| 4 \/ 5 | | 1 -2*\/ 5 | |2 -2 |
K2 = | | + | | + | |
| ___ | | ___ | |-2 -18|
|\/ 5 -18 | |-2*\/ 5 -18 | | ___ |
| 4 2 \/ 5 | | ___| | ___|
| | | 1 0 -2*\/ 5 | | 4 0 \/ 5 |
| ___| | | | |
K3 = | 2 1 -2*\/ 5 | + | 0 2 -2 | + | 0 2 -2 |
| | | | | |
| ___ ___ | | ___ | | ___ |
|\/ 5 -2*\/ 5 -18 | |-2*\/ 5 -2 -18 | |\/ 5 -2 -18 |
$$I_{1} = 7$$
$$I_{2} = 10$$
$$I_{3} = 0$$
$$I_{4} = -250$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 7 \lambda^{2} - 10 \lambda$$
$$K_{2} = -155$$
$$K_{3} = -375$$
Como
$$I_{3} = 0 \wedge I_{2} \neq 0 \wedge I_{4} \neq 0$$
entonces por razón de tipos de rectas:
hay que
Formulamos la ecuación característica para nuestra superficie:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
o
$$\lambda^{3} - 7 \lambda^{2} + 10 \lambda = 0$$
$$\lambda_{1} = 5$$
$$\lambda_{2} = 2$$
$$\lambda_{3} = 0$$
entonces la forma canónica de la ecuación será
$$\tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
y
$$- \tilde z 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
$$5 \tilde x^{2} + 2 \tilde y^{2} + 10 \tilde z = 0$$
y
$$5 \tilde x^{2} + 2 \tilde y^{2} - 10 \tilde z = 0$$
$$2 \tilde z + \left(\frac{\tilde x^{2}}{1} + \frac{\tilde y^{2}}{\frac{5}{2}}\right) = 0$$
y
$$- 2 \tilde z + \left(\frac{\tilde x^{2}}{1} + \frac{\tilde y^{2}}{\frac{5}{2}}\right) = 0$$
es la ecuación para el tipo paraboloide elíptico
- está reducida a la forma canónica