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polimi-cheatsheet
GitHub Repository: polimi-cheatsheet/GAL-cheatsheet
 Path: blob/master/esercizi/2.conica-centro.tex
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\subsection{Conica a centro}

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$\mathfrak{C}:\ 3x^2-2xy+3y^2-x-y-2=0$

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$

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A = \left[

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	\arraycolsep=2.0pt\def\arraystretch{1.0}

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	\begin{array}{@{}cc@{}}

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		3  & -1 \\

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		-1 & 3

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	\end{array}

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\right]

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\

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B = \left[

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	\arraycolsep=2.0pt\def\arraystretch{1.0}

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	\begin{array}{@{}ccc@{}}

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		3  & -1 & -1/2 \\

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		-1 & 3  & -1/2 \\

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		-1/2  & -1/2 & -2 \\

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	\end{array}

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\right]

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$

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\begin{tabular}{l}

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	$I_1 = \tr A = 6$ \\

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	$I_2 = \det A = 8$ \\ 

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	$I_3 = \det B = -18$

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\end{tabular}

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$I_2 \neq 0 \Rightarrow$ conica a centro. $I_1 I_3 < 0 \Rightarrow$ ellisse reale

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Calcolo il centro che soddisfa $A \vec{v} = -\vec{b}$

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$

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\left[

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	\arraycolsep=2.0pt\def\arraystretch{1.0}

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	\begin{array}{@{}cc|c@{}}

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		3  & -1 & 1/2 \\

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		-1 & 3  & 1/2

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	\end{array}

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\right]

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\rightarrow

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\left[

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	\arraycolsep=2.0pt\def\arraystretch{1.0}

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	\begin{array}{@{}cc|c@{}}

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		3  & -1   & 1/2 \\

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		0  & 8/3  & 2/3 

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	\end{array}

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\right]

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\Rightarrow

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O'_{|\Base} = \vec{v} =

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\begin{bmatrix}

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	1/4 \\[-0.3em]

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	1/4 \\

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\end{bmatrix}

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$

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%Calcolo autospazi di $A$ (autovettori sono paralleli assi di %$\mathfrak{C}$)

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$p(\lambda) = \det(A-\lambda I) = (2-\lambda)(4-\lambda)$

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$\ \Rightarrow \lambda_1 = 2, \lambda_2 = 4$

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$V_2 = \Lin 

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\begin{bmatrix}

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	1 \\[-0.3em]

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	1 \\

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\end{bmatrix}

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= \Lin

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\begin{bmatrix}

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	1/\sqrt{2} \\[-0.3em]

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	1/\sqrt{2} \\

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\end{bmatrix}

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\quad

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V_4 = \Lin 

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\begin{bmatrix}

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	-1 \\[-0.3em]

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	1 \\

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\end{bmatrix}

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= \Lin

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\begin{bmatrix}

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	-1/\sqrt{2} \\[-0.3em]

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	1/\sqrt{2} \\

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\end{bmatrix}

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$

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U ortogonale che diagonalizza A ($\det U = 1$ e $U$ ortonormale, ovvero descrive una rotazione)

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$

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\begin{bmatrix}

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	x \\[-0.3em]

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	y \\

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\end{bmatrix}

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= U

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\begin{bmatrix}

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	x' \\[-0.3em]

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	y' \\

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\end{bmatrix}

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+ \vec{v}=

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\left[

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	\arraycolsep=2.0pt\def\arraystretch{1.0}

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	\begin{array}{@{}cc@{}}

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		1/\sqrt{2} & -1/\sqrt{2} \\

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		1/\sqrt{2} & 1/\sqrt{2}

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	\end{array}

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\right]

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\begin{bmatrix}

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	x' \\[-0.3em]

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	y' \\

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\end{bmatrix}

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+

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\begin{bmatrix}

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	1/4 \\[-0.3em]

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	1/4 \\

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\end{bmatrix}

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$

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Essendo un ellisse: $\lambda_1x'^2 + \lambda_2y'^2 + c = 2x'^2 + 4y'^2 + c = 0$

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$

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B' = \left[

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	\arraycolsep=2.0pt\def\arraystretch{1.0}

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	\begin{array}{@{}ccc@{}}

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		2 & 0 & 0 \\

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		0 & 4 & 0 \\

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		0 & 0 & c \\

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	\end{array}

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\right]

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$

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\begin{tabular}{l}

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	$\Rightarrow I_3 = \det B' = 8c = \det B = -18$ \\

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	$\Rightarrow c = -9/4$

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\end{tabular}

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$\Rightarrow \mathfrak{C'}:\ 2x'^2 + 4y'^2 -9/4 = 0$

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$\Rightarrow \mathfrak{C'}:\ \frac{8}{9}x'^2 + \frac{16}{9}y'^2 = 1$

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