$\begin{array}{l} Given: | \begin{array}{ccc} a & b & a α + b \\ b & c & b α + c \\ a c + b & b c + c & 0 \end{array} | = 0 \\ Applying C_{3} \to C_{3} - (α C_{1} + C_{2}) \end{array}$ If the determinant $| \begin{array}{ccc} a & b & 0 \\ b & c & 0 \\ a α + b & b c + c & - (a c^{2} + 2 b c + c) \end{array} | = 0$ $\Rightarrow - (a α^{2} + 2 b α + c) (a c - b^{2}) = 0$ $\Rightarrow a α^{2} + 2 b α + c = 0 or b^{2} = a c$ $∴ α$ is root of $a x^{2} + 2 b x + c$ or $a, b, c$ are in GP.