Mister Exam
36x^2+16y^2+9z^2+72x-128y-36z+184=0 canonical form
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2 2 2 184 - 128*y - 36*z + 9*z + 16*y + 36*x + 72*x = 0
$$36 x^{2} + 72 x + 16 y^{2} - 128 y + 9 z^{2} - 36 z + 184 = 0$$
36*x^2 + 72*x + 16*y^2 - 128*y + 9*z^2 - 36*z + 184 = 0
Invariants method
Given equation of the surface of 2-order:
$$36 x^{2} + 72 x + 16 y^{2} - 128 y + 9 z^{2} - 36 z + 184 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0$$
where
$$a_{11} = 36$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{14} = 36$$
$$a_{22} = 16$$
$$a_{23} = 0$$
$$a_{24} = -64$$
$$a_{33} = 9$$
$$a_{34} = -18$$
$$a_{44} = 184$$
The invariants of the equation when converting coordinates are determinants:
$$I_{1} = a_{11} + a_{22} + a_{33}$$
$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}a_{11} & a_{12} & a_{13} & a_{14}\\a_{12} & a_{22} & a_{23} & a_{24}\\a_{13} & a_{23} & a_{33} & a_{34}\\a_{14} & a_{24} & a_{34} & a_{44}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12} & a_{13}\\a_{12} & a_{22} - \lambda & a_{23}\\a_{13} & a_{23} & a_{33} - \lambda\end{matrix}\right|$$
substitute coefficients
$$I_{1} = 61$$
$$I_{3} = \left|\begin{matrix}36 & 0 & 0\\0 & 16 & 0\\0 & 0 & 9\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}36 & 0 & 0 & 36\\0 & 16 & 0 & -64\\0 & 0 & 9 & -18\\36 & -64 & -18 & 184\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}36 - \lambda & 0 & 0\\0 & 16 - \lambda & 0\\0 & 0 & 9 - \lambda\end{matrix}\right|$$
$$I_{1} = 61$$
$$I_{2} = 1044$$
$$I_{3} = 5184$$
$$I_{4} = -746496$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 61 \lambda^{2} - 1044 \lambda + 5184$$
$$K_{2} = 5508$$
$$K_{3} = -41472$$
Because
then by type of surface:
you need to
Make the characteristic equation for the surface:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
or
$$\lambda^{3} - 61 \lambda^{2} + 1044 \lambda - 5184 = 0$$
$$\lambda_{1} = 36$$
$$\lambda_{2} = 16$$
$$\lambda_{3} = 9$$
then the canonical form of the equation will be
$$\left(\tilde z^{2} \lambda_{3} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right)\right) + \frac{I_{4}}{I_{3}} = 0$$
$$36 \tilde x^{2} + 16 \tilde y^{2} + 9 \tilde z^{2} - 144 = 0$$
this equation is fora type ellipsoid
- reduced to canonical form
$$36 x^{2} + 72 x + 16 y^{2} - 128 y + 9 z^{2} - 36 z + 184 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + 2 a_{14} x + a_{22} y^{2} + 2 a_{23} y z + 2 a_{24} y + a_{33} z^{2} + 2 a_{34} z + a_{44} = 0$$
where
$$a_{11} = 36$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{14} = 36$$
$$a_{22} = 16$$
$$a_{23} = 0$$
$$a_{24} = -64$$
$$a_{33} = 9$$
$$a_{34} = -18$$
$$a_{44} = 184$$
The invariants of the equation when converting coordinates are determinants:
$$I_{1} = a_{11} + a_{22} + a_{33}$$
|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|substitute coefficients
$$I_{1} = 61$$
|36 0 | |16 0| |36 0|
I2 = | | + | | + | |
|0 16| |0 9| |0 9|$$I_{3} = \left|\begin{matrix}36 & 0 & 0\\0 & 16 & 0\\0 & 0 & 9\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}36 & 0 & 0 & 36\\0 & 16 & 0 & -64\\0 & 0 & 9 & -18\\36 & -64 & -18 & 184\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}36 - \lambda & 0 & 0\\0 & 16 - \lambda & 0\\0 & 0 & 9 - \lambda\end{matrix}\right|$$
|36 36 | |16 -64| | 9 -18|
K2 = | | + | | + | |
|36 184| |-64 184| |-18 184| |36 0 36 | |16 0 -64| |36 0 36 |
| | | | | |
K3 = |0 16 -64| + | 0 9 -18| + |0 9 -18|
| | | | | |
|36 -64 184| |-64 -18 184| |36 -18 184|$$I_{1} = 61$$
$$I_{2} = 1044$$
$$I_{3} = 5184$$
$$I_{4} = -746496$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 61 \lambda^{2} - 1044 \lambda + 5184$$
$$K_{2} = 5508$$
$$K_{3} = -41472$$
Because
I3 != 0
then by type of surface:
you need to
Make the characteristic equation for the surface:
$$- I_{1} \lambda^{2} + I_{2} \lambda - I_{3} + \lambda^{3} = 0$$
or
$$\lambda^{3} - 61 \lambda^{2} + 1044 \lambda - 5184 = 0$$
$$\lambda_{1} = 36$$
$$\lambda_{2} = 16$$
$$\lambda_{3} = 9$$
then the canonical form of the equation will be
$$\left(\tilde z^{2} \lambda_{3} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right)\right) + \frac{I_{4}}{I_{3}} = 0$$
$$36 \tilde x^{2} + 16 \tilde y^{2} + 9 \tilde z^{2} - 144 = 0$$
2 2 2
\tilde x \tilde y \tilde z
--------- + --------- + --------- = 1
2 2 2
/ 1 \ / 1 \ / 1 \
|------| |------| |------|
\6*1/12/ \4*1/12/ \3*1/12/ this equation is fora type ellipsoid
- reduced to canonical form