Given equation of the surface of 2-order:
$$8 x^{2} - 4 x y + 4 x z + 7 y^{2} + 9 z^{2} = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x z + a_{22} y^{2} + 2 a_{23} y z + a_{33} z^{2} + 2 a_{14} x + 2 a_{24} y + 2 a_{34} z + a_{44} = 0$$
where
$$a_{11} = 8$$
$$a_{12} = -2$$
$$a_{13} = 2$$
$$a_{14} = 0$$
$$a_{22} = 7$$
$$a_{23} = 0$$
$$a_{24} = 0$$
$$a_{33} = 9$$
$$a_{34} = 0$$
$$a_{44} = 0$$
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
$$I_{1} = 24$$
|8 -2| |7 0| |8 2|
I2 = | | + | | + | |
|-2 7 | |0 9| |2 9|
$$I_{3} = \left|\begin{matrix}8 & -2 & 2\\-2 & 7 & 0\\2 & 0 & 9\end{matrix}\right|$$
$$I_{4} = \left|\begin{matrix}8 & -2 & 2 & 0\\-2 & 7 & 0 & 0\\2 & 0 & 9 & 0\\0 & 0 & 0 & 0\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda + 8 & -2 & 2\\-2 & - \lambda + 7 & 0\\2 & 0 & - \lambda + 9\end{matrix}\right|$$
|8 0| |7 0| |9 0|
K2 = | | + | | + | |
|0 0| |0 0| |0 0|
|8 -2 0| |7 0 0| |8 2 0|
| | | | | |
K3 = |-2 7 0| + |0 9 0| + |2 9 0|
| | | | | |
|0 0 0| |0 0 0| |0 0 0|
$$I_{1} = 24$$
$$I_{2} = 183$$
$$I_{3} = 440$$
$$I_{4} = 0$$
$$I{\left(\lambda \right)} = - \lambda^{3} + 24 \lambda^{2} - 183 \lambda + 440$$
$$K_{2} = 0$$
$$K_{3} = 0$$
Because
I3 != 0
then by type of surface:
you need to
Make the characteristic equation for the surface:
$$- I_{1} \lambda^{2} + \lambda^{3} + I_{2} \lambda - I_{3} = 0$$
or
$$\lambda^{3} - 24 \lambda^{2} + 183 \lambda - 440 = 0$$
Solve this equation$$\lambda_{1} = 11$$
$$\lambda_{2} = 8$$
$$\lambda_{3} = 5$$
then the canonical form of the equation will be
$$\left(\tilde z^{2} \lambda_{3} + \left(\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2}\right)\right) + \frac{I_{4}}{I_{3}} = 0$$
$$11 \tilde x^{2} + 8 \tilde y^{2} + 5 \tilde z^{2} = 0$$
$$\frac{\tilde z^{2}}{\left(\frac{\sqrt{5}}{5}\right)^{2}} + \left(\frac{\tilde x^{2}}{\left(\frac{\sqrt{11}}{11}\right)^{2}} + \frac{\tilde y^{2}}{\left(\frac{\sqrt{2}}{4}\right)^{2}}\right) = 0$$
this equation is fora type imaginary cone
- reduced to canonical form