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(B) Expanding the determinant $\Delta$ using column operations:Apply $C_3 	o C_3 - (\alpha C_1 + C_2)$:$\Delta = \begin{vmatrix} a & b & 0 \ b & c &

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$a, b, c$ are in geometric progression

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(B) Expanding the determinant $\Delta$ using column operations:Apply $C_3 	o C_3 - (\alpha C_1 + C_2)$:$\Delta = \begin{vmatrix} a & b & 0 \ b & c & 0 \ a\alpha + b & b\alpha + c & -(a\alpha^2 + 2b\alpha + c) \end{vmatrix}$Expanding along $C_3$:$\Delta = -(a\alpha^2 + 2b\alpha + c) \begin{vmatrix} a & b \ b & c \end{vmatrix}$$\Delta = -(a\alpha^2 + 2b\alpha + c)(ac - b^2)$$\Delta = (b^2 - ac)(a\alpha^2 + 2b\alpha + c)$For $\Delta = 0$,either $b^2 - ac = 0$ or $a\alpha^2 + 2b\alpha + c = 0$.If $b^2 - ac = 0$,then $b^2 = ac$,which means $a, b, c$ are in geometric progression.

If the determinant $\Delta = \begin{vmatrix} a & b & a\alpha + b \\ b & c & b\alpha + c \\ a\alpha + b & b\alpha + c & 0 \end{vmatrix} = 0$,then:

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