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Finding the critical line for a set of algebraic equations
I am trying to find a particular line where the determinant of the resulting quadratic form is zero, the line through $x=(0,0,0)$ and $y=(1,1,1)$. I am trying to do this in Java. The first step is to find a matrix representation of the form $(a + bx + cy)(dx+ey+f)$. In this case that is $$3x^2-2x+y^2-2y$$ However, the correct form would seem to be $$3x^2-2x+y^2-2y-x^2+4y-2x$$ but in this case the determinant is no longer 0 when $x=1$ and $y=1$.
How can I go about finding the correct critical line?
A:
Let $A = \begin{bmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 1 & 1\end{bmatrix}$, then $A = O + \begin{bmatrix} -1 & -1 & -1 \end{bmatrix}$. The $3 \times 3$ matrix $B = -2O$ is symmetric, hence has a characteristic polynomial of $\lambda^2 + \
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