Geometrie analitica si diferentiala

Conice cu centru

Fie conica de ecuatie

@d\underbrace{a_{11}x^2+2a_{12}xy+a_{22}y^2+2a_{13}x+2a_{23}y+a_{33}}_{f(x,y)}=0,@d

avand invariantii $$I=a_{11}+a_{22},\ \delta=\left|\begin{array}{cc}a_{11}&a_{12}\\a_{12}&a_{22}\end{array}\right|,\
\Delta=\left|\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{12}&a_{22}&a_{23}\\
a_{13}&a_{23}&a_{33}\end{array}\right|$$.

Daca $$\delta\ne0$$, atunci sistemul

@d\begin{cases}a_{11}x_0+a_{12}y_0+a_{13}=0\\ a_{12}x_0+a_{22}y_0+a_{23}=0\end{cases}@d

are solutie unica, iar dupa efectuarea translatiei de ecuatii $$\begin{cases}x=x_0+x'\\y=y_0+y'\end{cases}$$ ecuatia conicei in reperul cu originea in $$C(x_0,y_0)$$ devine

$$a_{11}x'^2+2a_{12}x'y'+a_{22}y'^2+f(x_0,y_0)=0$$

Prin calcul se obtine $$f(x_0,y_0)=\frac{\Delta}{\delta}$$, iar forma patratica determinata de primii 3 termeni are forma canonica $$\lambda_1X^2+\lambda_2Y^2$$, unde $$\lambda_1,\lambda_2$$ sunt valorile proprii ale matricei $$A=\left(\begin{array}{cc}a_{11}&a_{12}\\a_{12}&a_{22}\end{array}\right)$$, asadar conica are forma canonica

$$\lambda_1X^2+\lambda_2Y^2+\frac{\Delta}{\delta}=0$$,

unde $$\lambda_1,\lambda_2$$ sunt radacinile ecuatiei caracteristice

$$\lambda^2-I\lambda+\delta=0$$

alese astfel incat $$(\lambda_1-\lambda_2)a_{12}>0$$. Coordonatele $$X,Y$$ corespund unei rotatii de unghi $$\theta$$ dat prin

$$\operatorname{tg}\theta=\displaystyle\frac{\lambda_1-a_{11}}{a_{12}}$$,

deci legaturile dintre coordonatele initiale $$x,y$$ si cele in care avem forma canonica $$X,Y$$ sunt

$$\begin{cases}x=x_0+X\cos\theta-Y\sin\theta\\y=y_0+X\sin\theta+Y\cos\theta\end{cases}.$$

Last modified: Friday, 13 February 2015, 05:27 PM
Skip Navigation

Navigation