Let the system of linear equations
$x + y + α z = 2; 3 x + y + z = 4; x + 2 z = 1$
have a unique solution $(x, y, z)$ . If $(α, x), (y, α) a n d (x, - y)$ are collinear points, then the sum of absolute values of all possible values of $α$ is

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Detailed Solution

$\begin{array}{l} Δ = |\begin{array}{lll} 1 & 1 & α \\ 3 & 1 & 1 \\ 1 & 0 & 2 \end{array}| = - (α + 3), Δ_{1} = |\begin{array}{lll} 2 & 1 & α \\ 4 & 1 & 1 \\ 1 & 0 & 2 \end{array}| = - (3 + α) \\ \begin{array}{l} Δ_{2} = |\begin{array}{lll} 1 & 2 & α \\ 3 & 4 & 1 \\ 1 & 1 & 2 \end{array}| = - (α + 3), Δ_{3} = |\begin{array}{lll} 1 & 1 & 2 \\ 3 & 1 & 4 \\ 1 & 0 & 1 \end{array}| = 0 \\ α \neq - 3, x = 1, y = 1, z = 0, points (α, 1), (1, α) & (1, - 1) are collinear \\ |\begin{array}{ccc} α & 1 & 1 \\ 1 & α & 1 \\ 1 & - 1 & 1 \end{array}| = 0 \begin{array}{l} \Rightarrow α (α + 1) - 1 (1 - 1) + 1 (- 1 - α) = 0 \\ \Rightarrow α^{2} + α - 1 - α = 0 \\ \Rightarrow α = \pm 1 \end{array} \end{array} \end{array}$