Given line equation of 2-order:
$$9 x^{2} - 16 y^{2} + 18 x + 64 y - 199 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + a_{22} y^{2} + 2 a_{13} x + 2 a_{23} y + a_{33} = 0$$
where
$$a_{11} = 9$$
$$a_{12} = 0$$
$$a_{13} = 9$$
$$a_{22} = -16$$
$$a_{23} = 32$$
$$a_{33} = -199$$
The invariants of the equation when converting coordinates are determinants:
$$I_{1} = a_{11} + a_{22}$$
|a11 a12|
I2 = | |
|a12 a22|
$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12}\\a_{12} & a_{22} - \lambda\end{matrix}\right|$$
|a11 a13| |a22 a23|
K2 = | | + | |
|a13 a33| |a23 a33|
substitute coefficients
$$I_{1} = -7$$
|9 0 |
I2 = | |
|0 -16|
$$I_{3} = \left|\begin{matrix}9 & 0 & 9\\0 & -16 & 32\\9 & 32 & -199\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda + 9 & 0\\0 & - \lambda - 16\end{matrix}\right|$$
|9 9 | |-16 32 |
K2 = | | + | |
|9 -199| |32 -199|
$$I_{1} = -7$$
$$I_{2} = -144$$
$$I_{3} = 20736$$
$$I{\left(\lambda \right)} = \lambda^{2} + 7 \lambda - 144$$
$$K_{2} = 288$$
Because
$$I_{2} < 0 \wedge I_{3} \neq 0$$
then by line type:
this equation is of type : hyperbola
Make the characteristic equation for the line:
$$- I_{1} \lambda + \lambda^{2} + I_{2} = 0$$
or
$$\lambda^{2} + 7 \lambda - 144 = 0$$
Solve this equation$$\lambda_{1} = 9$$
$$\lambda_{2} = -16$$
then the canonical form of the equation will be
$$\tilde x^{2} \lambda_{1} + \tilde y^{2} \lambda_{2} + \frac{I_{3}}{I_{2}} = 0$$
or
$$9 \tilde x^{2} - 16 \tilde y^{2} - 144 = 0$$
$$\frac{\tilde x^{2}}{16} - \frac{\tilde y^{2}}{9} = 1$$
- reduced to canonical form