Consider the conic bundle: $\mathcal{C_{\lambda,\mu}} : \lambda(4x_0^2+3x_1^2-4x_ox_2-2x_1x_2)+\mu(4x_0^2+5x_1^2+4x_0x_1-8x_0x_2-4x_1x_2)=0$
Find the locus of the centers of those conics and write the equation.
How to find it? I did this, but I think I was following the wrong way:
The matrix which represents the bundle is:
$ \begin{bmatrix} 4\lambda+4\mu & 2\mu & -2\lambda-4\mu \\ 2\mu & 3\lambda+5\mu & -\lambda-2\mu \\ -2\lambda-4\mu & -\lambda-2\mu & 0 \\ \end{bmatrix} $
from the theory I know that I can find the center of a conic solving this system:
$ \begin{bmatrix} c_{11} & c_{12} \\ c_{12} & c_{22} \\ \end{bmatrix} $ $ \begin{bmatrix} x \\ y \\ \end{bmatrix} $ = $ \begin{bmatrix} -c_{01} \\ -c_{02} \\ \end{bmatrix} $ replacing: $ \begin{bmatrix} 3\lambda+5\mu & -\lambda-2\mu \\ -\lambda-2\mu & 0 \\ \end{bmatrix} $ $ \begin{bmatrix} x \\ y \\ \end{bmatrix} $ = $ \begin{bmatrix} -2\mu \\ 2\lambda+4\mu \\ \end{bmatrix} $
then $ \left[ \begin{array}{cc|c} 3\lambda+5\mu&-\lambda-2\mu&-2\mu\\ -\lambda-2\mu&0&2\lambda+4\mu \end{array} \right] $ , after passages: $ \left[ \begin{array}{cc|c} 1&0&-2\\ 0&1&-2(\lambda+3\mu) \end{array} \right] $.
So the coordinates of the center should be $[1, -2, -2(\lambda+3\mu)]$. Is this right? There are other ways? Thank you