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How to use it?
- Canonical form:
- 9x²-81y²=729
- 2x^2+6y^2+2z^2+8xz
- sin^2x+3cos^2x
- y-x
- Plot:
- z=3/sqrt(x^2+y-9)
-
sin(x^e)+cos(y^e)=1
-
x^2+4y^2
- tan(sin(e−1x))−x=0
-
Identical expressions
- 9x²-81y²= seven hundred and twenty-nine
- 9x² minus 81y² equally 729
- 9x² minus 81y² equally seven hundred and twenty minus nine
-
Similar expressions
- (7^2)*(x^2)+((-1)^6)*(6^2)*(y^2)-2*(7^2)*9x+(7^2)*((9^2)-(6^2))
- 9x²+81y²=729
9x²-81y²=729 canonical form
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The solution
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2 2 -729 - 81*y + 9*x = 0
$$9 x^{2} - 81 y^{2} - 729 = 0$$
9*x^2 - 81*y^2 - 729 = 0
Detail solution
Given line equation of 2-order:
$$9 x^{2} - 81 y^{2} - 729 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x + a_{22} y^{2} + 2 a_{23} y + a_{33} = 0$$
where
$$a_{11} = 9$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{22} = -81$$
$$a_{23} = 0$$
$$a_{33} = -729$$
To calculate the determinant
$$\Delta = \left|\begin{matrix}a_{11} & a_{12}\\a_{12} & a_{22}\end{matrix}\right|$$
or, substitute
$$\Delta = \left|\begin{matrix}9 & 0\\0 & -81\end{matrix}\right|$$
$$\Delta = -729$$
Because
$$\Delta$$
is not equal to 0, then
find the center of the canonical coordinate system. To do it, solve the system of equations
$$a_{11} x_{0} + a_{12} y_{0} + a_{13} = 0$$
$$a_{12} x_{0} + a_{22} y_{0} + a_{23} = 0$$
substitute coefficients
$$9 x_{0} = 0$$
$$- 81 y_{0} = 0$$
then
$$x_{0} = 0$$
$$y_{0} = 0$$
Thus, we have the equation in the coordinate system O'x'y'
$$a'_{33} + a_{11} x'^{2} + 2 a_{12} x' y' + a_{22} y'^{2} = 0$$
where
$$a'_{33} = a_{13} x_{0} + a_{23} y_{0} + a_{33}$$
or
$$a'_{33} = -729$$
$$a'_{33} = -729$$
then equation turns into
$$9 x'^{2} - 81 y'^{2} - 729 = 0$$
Given equation is hyperbole
$$\frac{\tilde x^{2}}{81} - \frac{\tilde y^{2}}{9} = 1$$
- reduced to canonical form
The center of canonical coordinate system at point O
Basis of the canonical coordinate system
$$\vec e_1 = \left( 1, \ 0\right)$$
$$\vec e_2 = \left( 0, \ 1\right)$$
$$9 x^{2} - 81 y^{2} - 729 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x + a_{22} y^{2} + 2 a_{23} y + a_{33} = 0$$
where
$$a_{11} = 9$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{22} = -81$$
$$a_{23} = 0$$
$$a_{33} = -729$$
To calculate the determinant
$$\Delta = \left|\begin{matrix}a_{11} & a_{12}\\a_{12} & a_{22}\end{matrix}\right|$$
or, substitute
$$\Delta = \left|\begin{matrix}9 & 0\\0 & -81\end{matrix}\right|$$
$$\Delta = -729$$
Because
$$\Delta$$
is not equal to 0, then
find the center of the canonical coordinate system. To do it, solve the system of equations
$$a_{11} x_{0} + a_{12} y_{0} + a_{13} = 0$$
$$a_{12} x_{0} + a_{22} y_{0} + a_{23} = 0$$
substitute coefficients
$$9 x_{0} = 0$$
$$- 81 y_{0} = 0$$
then
$$x_{0} = 0$$
$$y_{0} = 0$$
Thus, we have the equation in the coordinate system O'x'y'
$$a'_{33} + a_{11} x'^{2} + 2 a_{12} x' y' + a_{22} y'^{2} = 0$$
where
$$a'_{33} = a_{13} x_{0} + a_{23} y_{0} + a_{33}$$
or
$$a'_{33} = -729$$
$$a'_{33} = -729$$
then equation turns into
$$9 x'^{2} - 81 y'^{2} - 729 = 0$$
Given equation is hyperbole
$$\frac{\tilde x^{2}}{81} - \frac{\tilde y^{2}}{9} = 1$$
- reduced to canonical form
The center of canonical coordinate system at point O
(0, 0)
Basis of the canonical coordinate system
$$\vec e_1 = \left( 1, \ 0\right)$$
$$\vec e_2 = \left( 0, \ 1\right)$$
Invariants method
Given line equation of 2-order:
$$9 x^{2} - 81 y^{2} - 729 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x + a_{22} y^{2} + 2 a_{23} y + a_{33} = 0$$
where
$$a_{11} = 9$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{22} = -81$$
$$a_{23} = 0$$
$$a_{33} = -729$$
The invariants of the equation when converting coordinates are determinants:
$$I_{1} = a_{11} + a_{22}$$
$$I_{3} = \left|\begin{matrix}a_{11} & a_{12} & a_{13}\\a_{12} & a_{22} & a_{23}\\a_{13} & a_{23} & a_{33}\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}a_{11} - \lambda & a_{12}\\a_{12} & a_{22} - \lambda\end{matrix}\right|$$
substitute coefficients
$$I_{1} = -72$$
$$I_{3} = \left|\begin{matrix}9 & 0 & 0\\0 & -81 & 0\\0 & 0 & -729\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}9 - \lambda & 0\\0 & - \lambda - 81\end{matrix}\right|$$
$$I_{1} = -72$$
$$I_{2} = -729$$
$$I_{3} = 531441$$
$$I{\left(\lambda \right)} = \lambda^{2} + 72 \lambda - 729$$
$$K_{2} = 52488$$
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:
$$- I_{1} \lambda + I_{2} + \lambda^{2} = 0$$
or
$$\lambda^{2} + 72 \lambda - 729 = 0$$
$$\lambda_{1} = 9$$
$$\lambda_{2} = -81$$
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
$$9 \tilde x^{2} - 81 \tilde y^{2} - 729 = 0$$
$$\frac{\tilde x^{2}}{81} - \frac{\tilde y^{2}}{9} = 1$$
- reduced to canonical form
$$9 x^{2} - 81 y^{2} - 729 = 0$$
This equation looks like:
$$a_{11} x^{2} + 2 a_{12} x y + 2 a_{13} x + a_{22} y^{2} + 2 a_{23} y + a_{33} = 0$$
where
$$a_{11} = 9$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{22} = -81$$
$$a_{23} = 0$$
$$a_{33} = -729$$
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
$$I_{1} = -72$$
|9 0 |
I2 = | |
|0 -81|$$I_{3} = \left|\begin{matrix}9 & 0 & 0\\0 & -81 & 0\\0 & 0 & -729\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}9 - \lambda & 0\\0 & - \lambda - 81\end{matrix}\right|$$
|9 0 | |-81 0 |
K2 = | | + | |
|0 -729| | 0 -729|$$I_{1} = -72$$
$$I_{2} = -729$$
$$I_{3} = 531441$$
$$I{\left(\lambda \right)} = \lambda^{2} + 72 \lambda - 729$$
$$K_{2} = 52488$$
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:
$$- I_{1} \lambda + I_{2} + \lambda^{2} = 0$$
or
$$\lambda^{2} + 72 \lambda - 729 = 0$$
$$\lambda_{1} = 9$$
$$\lambda_{2} = -81$$
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
$$9 \tilde x^{2} - 81 \tilde y^{2} - 729 = 0$$
$$\frac{\tilde x^{2}}{81} - \frac{\tilde y^{2}}{9} = 1$$
- reduced to canonical form