Mister Exam
7x^2-6xy-y^2+26x-2y+7=0 canonical form
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2 2 7 - y - 2*y + 7*x + 26*x - 6*x*y = 0
$$7 x^{2} - 6 x y + 26 x - y^{2} - 2 y + 7 = 0$$
7*x^2 - 6*x*y + 26*x - y^2 - 2*y + 7 = 0
Detail solution
Given line equation of 2-order:
$$7 x^{2} - 6 x y + 26 x - y^{2} - 2 y + 7 = 0$$
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
$$a_{11} = 7$$
$$a_{12} = -3$$
$$a_{13} = 13$$
$$a_{22} = -1$$
$$a_{23} = -1$$
$$a_{33} = 7$$
To calculate the determinant
$$\Delta = \left|\begin{matrix}a_{11} & a_{12}\\a_{12} & a_{22}\end{matrix}\right|$$
or, substitute
$$\Delta = \left|\begin{matrix}7 & -3\\-3 & -1\end{matrix}\right|$$
$$\Delta = -16$$
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
$$7 x_{0} - 3 y_{0} + 13 = 0$$
$$- 3 x_{0} - y_{0} - 1 = 0$$
then
$$x_{0} = -1$$
$$y_{0} = 2$$
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
$$a'_{33} = 13 x_{0} - y_{0} + 7$$
$$a'_{33} = -8$$
then equation turns into
$$7 x'^{2} - 6 x' y' - y'^{2} - 8 = 0$$
Rotate the resulting coordinate system by an angle φ
$$x' = \tilde x \cos{\left(\phi \right)} - \tilde y \sin{\left(\phi \right)}$$
$$y' = \tilde x \sin{\left(\phi \right)} + \tilde y \cos{\left(\phi \right)}$$
φ - determined from the formula
$$\cot{\left(2 \phi \right)} = \frac{a_{11} - a_{22}}{2 a_{12}}$$
substitute coefficients
$$\cot{\left(2 \phi \right)} = - \frac{4}{3}$$
then
$$\phi = - \frac{\operatorname{acot}{\left(\frac{4}{3} \right)}}{2}$$
$$\sin{\left(2 \phi \right)} = - \frac{3}{5}$$
$$\cos{\left(2 \phi \right)} = \frac{4}{5}$$
$$\cos{\left(\phi \right)} = \sqrt{\frac{\cos{\left(2 \phi \right)}}{2} + \frac{1}{2}}$$
$$\sin{\left(\phi \right)} = \sqrt{1 - \cos^{2}{\left(\phi \right)}}$$
$$\cos{\left(\phi \right)} = \frac{3 \sqrt{10}}{10}$$
$$\sin{\left(\phi \right)} = - \frac{\sqrt{10}}{10}$$
substitute coefficients
$$x' = \frac{3 \sqrt{10} \tilde x}{10} + \frac{\sqrt{10} \tilde y}{10}$$
$$y' = - \frac{\sqrt{10} \tilde x}{10} + \frac{3 \sqrt{10} \tilde y}{10}$$
then the equation turns from
$$7 x'^{2} - 6 x' y' - y'^{2} - 8 = 0$$
to
$$- \left(- \frac{\sqrt{10} \tilde x}{10} + \frac{3 \sqrt{10} \tilde y}{10}\right)^{2} - 6 \left(- \frac{\sqrt{10} \tilde x}{10} + \frac{3 \sqrt{10} \tilde y}{10}\right) \left(\frac{3 \sqrt{10} \tilde x}{10} + \frac{\sqrt{10} \tilde y}{10}\right) + 7 \left(\frac{3 \sqrt{10} \tilde x}{10} + \frac{\sqrt{10} \tilde y}{10}\right)^{2} - 8 = 0$$
simplify
$$8 \tilde x^{2} - 2 \tilde y^{2} - 8 = 0$$
$$- 8 \tilde x^{2} + 2 \tilde y^{2} + 8 = 0$$
Given equation is hyperbole
$$\frac{\tilde x^{2}}{1} - \frac{\tilde y^{2}}{4} = 1$$
- reduced to canonical form
The center of canonical coordinate system at point O
Basis of the canonical coordinate system
$$\vec e_1 = \left( \frac{3 \sqrt{10}}{10}, \ - \frac{\sqrt{10}}{10}\right)$$
$$\vec e_2 = \left( \frac{\sqrt{10}}{10}, \ \frac{3 \sqrt{10}}{10}\right)$$
$$7 x^{2} - 6 x y + 26 x - y^{2} - 2 y + 7 = 0$$
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
$$a_{11} = 7$$
$$a_{12} = -3$$
$$a_{13} = 13$$
$$a_{22} = -1$$
$$a_{23} = -1$$
$$a_{33} = 7$$
To calculate the determinant
$$\Delta = \left|\begin{matrix}a_{11} & a_{12}\\a_{12} & a_{22}\end{matrix}\right|$$
or, substitute
$$\Delta = \left|\begin{matrix}7 & -3\\-3 & -1\end{matrix}\right|$$
$$\Delta = -16$$
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
$$7 x_{0} - 3 y_{0} + 13 = 0$$
$$- 3 x_{0} - y_{0} - 1 = 0$$
then
$$x_{0} = -1$$
$$y_{0} = 2$$
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
$$a'_{33} = 13 x_{0} - y_{0} + 7$$
$$a'_{33} = -8$$
then equation turns into
$$7 x'^{2} - 6 x' y' - y'^{2} - 8 = 0$$
Rotate the resulting coordinate system by an angle φ
$$x' = \tilde x \cos{\left(\phi \right)} - \tilde y \sin{\left(\phi \right)}$$
$$y' = \tilde x \sin{\left(\phi \right)} + \tilde y \cos{\left(\phi \right)}$$
φ - determined from the formula
$$\cot{\left(2 \phi \right)} = \frac{a_{11} - a_{22}}{2 a_{12}}$$
substitute coefficients
$$\cot{\left(2 \phi \right)} = - \frac{4}{3}$$
then
$$\phi = - \frac{\operatorname{acot}{\left(\frac{4}{3} \right)}}{2}$$
$$\sin{\left(2 \phi \right)} = - \frac{3}{5}$$
$$\cos{\left(2 \phi \right)} = \frac{4}{5}$$
$$\cos{\left(\phi \right)} = \sqrt{\frac{\cos{\left(2 \phi \right)}}{2} + \frac{1}{2}}$$
$$\sin{\left(\phi \right)} = \sqrt{1 - \cos^{2}{\left(\phi \right)}}$$
$$\cos{\left(\phi \right)} = \frac{3 \sqrt{10}}{10}$$
$$\sin{\left(\phi \right)} = - \frac{\sqrt{10}}{10}$$
substitute coefficients
$$x' = \frac{3 \sqrt{10} \tilde x}{10} + \frac{\sqrt{10} \tilde y}{10}$$
$$y' = - \frac{\sqrt{10} \tilde x}{10} + \frac{3 \sqrt{10} \tilde y}{10}$$
then the equation turns from
$$7 x'^{2} - 6 x' y' - y'^{2} - 8 = 0$$
to
$$- \left(- \frac{\sqrt{10} \tilde x}{10} + \frac{3 \sqrt{10} \tilde y}{10}\right)^{2} - 6 \left(- \frac{\sqrt{10} \tilde x}{10} + \frac{3 \sqrt{10} \tilde y}{10}\right) \left(\frac{3 \sqrt{10} \tilde x}{10} + \frac{\sqrt{10} \tilde y}{10}\right) + 7 \left(\frac{3 \sqrt{10} \tilde x}{10} + \frac{\sqrt{10} \tilde y}{10}\right)^{2} - 8 = 0$$
simplify
$$8 \tilde x^{2} - 2 \tilde y^{2} - 8 = 0$$
$$- 8 \tilde x^{2} + 2 \tilde y^{2} + 8 = 0$$
Given equation is hyperbole
$$\frac{\tilde x^{2}}{1} - \frac{\tilde y^{2}}{4} = 1$$
- reduced to canonical form
The center of canonical coordinate system at point O
(-1, 2)
Basis of the canonical coordinate system
$$\vec e_1 = \left( \frac{3 \sqrt{10}}{10}, \ - \frac{\sqrt{10}}{10}\right)$$
$$\vec e_2 = \left( \frac{\sqrt{10}}{10}, \ \frac{3 \sqrt{10}}{10}\right)$$
Invariants method
Given line equation of 2-order:
$$7 x^{2} - 6 x y + 26 x - y^{2} - 2 y + 7 = 0$$
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
$$a_{11} = 7$$
$$a_{12} = -3$$
$$a_{13} = 13$$
$$a_{22} = -1$$
$$a_{23} = -1$$
$$a_{33} = 7$$
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
$$I_{1} = 6$$
$$I_{3} = \left|\begin{matrix}7 & -3 & 13\\-3 & -1 & -1\\13 & -1 & 7\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}7 - \lambda & -3\\-3 & - \lambda - 1\end{matrix}\right|$$
$$I_{1} = 6$$
$$I_{2} = -16$$
$$I_{3} = 128$$
$$I{\left(\lambda \right)} = \lambda^{2} - 6 \lambda - 16$$
$$K_{2} = -128$$
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
$$\lambda^{2} - 6 \lambda - 16 = 0$$
$$\lambda_{1} = 8$$
$$\lambda_{2} = -2$$
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
$$8 \tilde x^{2} - 2 \tilde y^{2} - 8 = 0$$
$$\frac{\tilde x^{2}}{1} - \frac{\tilde y^{2}}{4} = 1$$
- reduced to canonical form
$$7 x^{2} - 6 x y + 26 x - y^{2} - 2 y + 7 = 0$$
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
$$a_{11} = 7$$
$$a_{12} = -3$$
$$a_{13} = 13$$
$$a_{22} = -1$$
$$a_{23} = -1$$
$$a_{33} = 7$$
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
$$I_{1} = 6$$
|7 -3|
I2 = | |
|-3 -1|$$I_{3} = \left|\begin{matrix}7 & -3 & 13\\-3 & -1 & -1\\13 & -1 & 7\end{matrix}\right|$$
$$I{\left(\lambda \right)} = \left|\begin{matrix}7 - \lambda & -3\\-3 & - \lambda - 1\end{matrix}\right|$$
|7 13| |-1 -1|
K2 = | | + | |
|13 7 | |-1 7 |$$I_{1} = 6$$
$$I_{2} = -16$$
$$I_{3} = 128$$
$$I{\left(\lambda \right)} = \lambda^{2} - 6 \lambda - 16$$
$$K_{2} = -128$$
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
$$\lambda^{2} - 6 \lambda - 16 = 0$$
$$\lambda_{1} = 8$$
$$\lambda_{2} = -2$$
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
$$8 \tilde x^{2} - 2 \tilde y^{2} - 8 = 0$$
$$\frac{\tilde x^{2}}{1} - \frac{\tilde y^{2}}{4} = 1$$
- reduced to canonical form