Sr Examen
x^2+y^2-ax+by+c=0 forma canónica
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Solución
Ha introducido
[src]
2 2 c + x + y + b*y - a*x = 0
$$- a x + b y + c + x^{2} + y^{2} = 0$$
-a*x + b*y + c + x^2 + y^2 = 0
Método de invariantes
Se da la ecuación de superficie de 2 grado:
$$- a x + b y + c + x^{2} + y^{2} = 0$$
Esta ecuación tiene la forma:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} c x + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} c y + 2 a_{24} y + a_{33} c^{2} + 2 a_{34} c + a_{44} = 0$$
donde
$$a_{11} = 1$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{14} = - \frac{a}{2}$$
$$a_{22} = 1$$
$$a_{23} = 0$$
$$a_{24} = \frac{b}{2}$$
$$a_{33} = 0$$
$$a_{34} = \frac{1}{2}$$
$$a_{44} = 0$$
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
$$I_{1} = 2$$
$$I_{3} = \left|\begin{matrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 0\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}1 & 0 & 0 & - \frac{a}{2}\\0 & 1 & 0 & \frac{b}{2}\\0 & 0 & 0 & \frac{1}{2}\\- \frac{a}{2} & \frac{b}{2} & \frac{1}{2} & 0\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}1 - \lambda & 0 & 0\\0 & 1 - \lambda & 0\\0 & 0 & - \lambda\end{matrix}\right|$$
$$I_{1} = 2$$
$$I_{2} = 1$$
$$I_{3} = 0$$
$$I_{4} = - \frac{1}{4}$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 2 \lambda^{2} - \lambda$$
$$K_{2} = - \frac{a^{2}}{4} - \frac{b^{2}}{4} - \frac{1}{4}$$
$$K_{3} = - \frac{a^{2}}{4} - \frac{b^{2}}{4} - \frac{1}{2}$$
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
$$\lambda^{3} - 2 \lambda^{2} + \lambda = 0$$
$$\lambda_{1} = 1$$
$$\lambda_{2} = 1$$
$$\lambda_{3} = 0$$
entonces la forma canónica de la ecuación será
$$\tilde c 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
y
$$- \tilde c 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
$$\tilde c + \tilde x^{2} + \tilde y^{2} = 0$$
y
$$- \tilde c + \tilde x^{2} + \tilde y^{2} = 0$$
y
es la ecuación para el tipo paraboloide elíptico
- está reducida a la forma canónica
$$- a x + b y + c + x^{2} + y^{2} = 0$$
Esta ecuación tiene la forma:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} c x + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} c y + 2 a_{24} y + a_{33} c^{2} + 2 a_{34} c + a_{44} = 0$$
donde
$$a_{11} = 1$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{14} = - \frac{a}{2}$$
$$a_{22} = 1$$
$$a_{23} = 0$$
$$a_{24} = \frac{b}{2}$$
$$a_{33} = 0$$
$$a_{34} = \frac{1}{2}$$
$$a_{44} = 0$$
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
$$I_{1} = 2$$
|1 0| |1 0| |1 0|
I2 = | | + | | + | |
|0 1| |0 0| |0 0|$$I_{3} = \left|\begin{matrix}1 & 0 & 0\\0 & 1 & 0\\0 & 0 & 0\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}1 & 0 & 0 & - \frac{a}{2}\\0 & 1 & 0 & \frac{b}{2}\\0 & 0 & 0 & \frac{1}{2}\\- \frac{a}{2} & \frac{b}{2} & \frac{1}{2} & 0\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}1 - \lambda & 0 & 0\\0 & 1 - \lambda & 0\\0 & 0 & - \lambda\end{matrix}\right|$$
| -a | | b|
| 1 ---| |1 -|
| 2 | | 2| | 0 1/2|
K2 = | | + | | + | |
|-a | |b | |1/2 0 |
|--- 0 | |- 0|
| 2 | |2 | | -a |
| 1 0 ---| | b | | -a |
| 2 | |1 0 - | | 1 0 ---|
| | | 2 | | 2 |
| b | | | | |
K3 = | 0 1 - | + |0 0 1/2| + | 0 0 1/2|
| 2 | | | | |
| | |b | |-a |
|-a b | |- 1/2 0 | |--- 1/2 0 |
|--- - 0 | |2 | | 2 |
| 2 2 | $$I_{1} = 2$$
$$I_{2} = 1$$
$$I_{3} = 0$$
$$I_{4} = - \frac{1}{4}$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 2 \lambda^{2} - \lambda$$
$$K_{2} = - \frac{a^{2}}{4} - \frac{b^{2}}{4} - \frac{1}{4}$$
$$K_{3} = - \frac{a^{2}}{4} - \frac{b^{2}}{4} - \frac{1}{2}$$
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
$$\lambda^{3} - 2 \lambda^{2} + \lambda = 0$$
$$\lambda_{1} = 1$$
$$\lambda_{2} = 1$$
$$\lambda_{3} = 0$$
entonces la forma canónica de la ecuación será
$$\tilde c 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
y
$$- \tilde c 2 \sqrt{\frac{\left(-1\right) I_{4}}{I_{2}}} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right) = 0$$
$$\tilde c + \tilde x^{2} + \tilde y^{2} = 0$$
y
$$- \tilde c + \tilde x^{2} + \tilde y^{2} = 0$$
2 2 \tilde x \tilde y --------- + --------- + 2*\tilde c = 0 1/2 1/2
y
2 2 \tilde x \tilde y --------- + --------- - 2*\tilde c = 0 1/2 1/2
es la ecuación para el tipo paraboloide elíptico
- está reducida a la forma canónica