Given line equation of 2-order:
$$9 x^{2} - 25 y^{2} - 18 x - 100 y - 316 = 0$$
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
$$a_{11} = 9$$
$$a_{12} = 0$$
$$a_{13} = -9$$
$$a_{22} = -25$$
$$a_{23} = -50$$
$$a_{33} = -316$$
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
$$I_{1} = -16$$
|9 0 |
I2 = | |
|0 -25|
$$I_{3} = \left|\begin{matrix}9 & 0 & -9\\0 & -25 & -50\\-9 & -50 & -316\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda + 9 & 0\\0 & - \lambda - 25\end{matrix}\right|$$
|9 -9 | |-25 -50 |
K2 = | | + | |
|-9 -316| |-50 -316|
$$I_{1} = -16$$
$$I_{2} = -225$$
$$I_{3} = 50625$$
$$I{\left(\lambda \right)} = \lambda^{2} + 16 \lambda - 225$$
$$K_{2} = 2475$$
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
$$\lambda^{2} + 16 \lambda - 225 = 0$$
Solve this equation$$\lambda_{1} = 9$$
$$\lambda_{2} = -25$$
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
$$9 \tilde x^{2} - 25 \tilde y^{2} - 225 = 0$$
$$\frac{\tilde x^{2}}{25} - \frac{\tilde y^{2}}{9} = 1$$
- reduced to canonical form