Mister Exam

x1^2+4x2^2 canonical form

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The solution

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  2       2    
x1  + 4*x2  = 0
$$x_{1}^{2} + 4 x_{2}^{2} = 0$$
x1^2 + 4*x2^2 = 0
Detail solution
Given line equation of 2-order:
$$x_{1}^{2} + 4 x_{2}^{2} = 0$$
This equation looks like:
$$a_{11} x_{2}^{2} + 2 a_{12} x_{1} x_{2} + a_{22} x_{1}^{2} + 2 a_{13} x_{2} + 2 a_{23} x_{1} + a_{33} = 0$$
where
$$a_{11} = 4$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{22} = 1$$
$$a_{23} = 0$$
$$a_{33} = 0$$
To calculate the determinant
$$\Delta = \left|\begin{matrix}a_{11} & a_{12}\\a_{12} & a_{22}\end{matrix}\right|$$
or, substitute
$$\Delta = \left|\begin{matrix}4 & 0\\0 & 1\end{matrix}\right|$$
$$\Delta = 4$$
Because
$$\Delta$$
is not equal to 0, then
find the center of the canonical coordinate system. To do it, solve the system of equations
$$a_{11} x_{20} + a_{12} x_{10} + a_{13} = 0$$
$$a_{12} x_{20} + a_{22} x_{10} + a_{23} = 0$$
substitute coefficients
$$4 x_{20} = 0$$
$$x_{10} = 0$$
then
$$x_{20} = 0$$
$$x_{10} = 0$$
Thus, we have the equation in the coordinate system O'x'y'
$$a_{11} x2'^{2} + 2 a_{12} x1' x2' + a_{22} x1'^{2} + a'_{33} = 0$$
where
$$a'_{33} = a_{13} x_{20} + a_{23} x_{10} + a_{33}$$
or
$$a'_{33} = 0$$
$$a'_{33} = 0$$
then The equation is transformed to
$$x1'^{2} + 4 x2'^{2} = 0$$
Given equation is degenerate ellipse
$$\frac{\tilde x1^{2}}{1^{2}} + \frac{\tilde x2^{2}}{\left(\frac{1}{2}\right)^{2}} = 0$$
- 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:
$$x_{1}^{2} + 4 x_{2}^{2} = 0$$
This equation looks like:
$$a_{11} x_{2}^{2} + 2 a_{12} x_{1} x_{2} + a_{22} x_{1}^{2} + 2 a_{13} x_{2} + 2 a_{23} x_{1} + a_{33} = 0$$
where
$$a_{11} = 4$$
$$a_{12} = 0$$
$$a_{13} = 0$$
$$a_{22} = 1$$
$$a_{23} = 0$$
$$a_{33} = 0$$
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
$$I_{1} = 5$$
     |4  0|
I2 = |    |
     |0  1|

$$I_{3} = \left|\begin{matrix}4 & 0 & 0\\0 & 1 & 0\\0 & 0 & 0\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}- \lambda + 4 & 0\\0 & - \lambda + 1\end{matrix}\right|$$
     |4  0|   |1  0|
K2 = |    | + |    |
     |0  0|   |0  0|

$$I_{1} = 5$$
$$I_{2} = 4$$
$$I_{3} = 0$$
$$I{\left(\lambda \right)} = \lambda^{2} - 5 \lambda + 4$$
$$K_{2} = 0$$
Because
$$I_{3} = 0 \wedge I_{2} > 0$$
then by line type:
this equation is of type : degenerate ellipse
Make the characteristic equation for the line:
$$- I_{1} \lambda + \lambda^{2} + I_{2} = 0$$
or
$$\lambda^{2} - 5 \lambda + 4 = 0$$
Solve this equation
$$\lambda_{1} = 4$$
$$\lambda_{2} = 1$$
then the canonical form of the equation will be
$$\tilde x1^{2} \lambda_{2} + \tilde x2^{2} \lambda_{1} + \frac{I_{3}}{I_{2}} = 0$$
or
$$\tilde x1^{2} + 4 \tilde x2^{2} = 0$$
$$\frac{\tilde x1^{2}}{1^{2}} + \frac{\tilde x2^{2}}{\left(\frac{1}{2}\right)^{2}} = 0$$
- reduced to canonical form
The graph
x1^2+4x2^2 canonical form