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(B) Let $\Delta = \left| \begin{array}{ccc} a & b & a\alpha + b \ b & c & b\alpha + c \ a\alpha + b & b\alpha + c & 0 \end{array} ight|$.Applying

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$G. P.$

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(B) Let $\Delta = \left| \begin{array}{ccc} a & b & a\alpha + b \ b & c & b\alpha + c \ a\alpha + b & b\alpha + c & 0 \end{array} ight|$.Applying the row operation $R_3 	o R_3 - \alpha R_1 - R_2$:$\Delta = \left| \begin{array}{ccc} a & b & a\alpha + b \ b & c & b\alpha + c \ 0 & 0 & -(a\alpha^2 + 2b\alpha + c) \end{array} ight|$.Expanding along the third row $(R_3)$:$\Delta = -(a\alpha^2 + 2b\alpha + c) \cdot \left| \begin{array}{cc} a & b \ b & c \end{array} ight| = -(a\alpha^2 + 2b\alpha + c)(ac - b^2) = (b^2 - ac)(a\alpha^2 + 2b\alpha + c)$.For $\Delta = 0$,we must have $b^2 - ac = 0$ or $a\alpha^2 + 2b\alpha + c = 0$.The condition $b^2 - ac = 0$ implies $b^2 = ac$,which means $a, b, c$ are in $G.P.$

The determinant $\left| \begin{array}{ccc} a & b & a\alpha + b \\ b & c & b\alpha + c \\ a\alpha + b & b\alpha + c & 0 \end{array} \right| = 0$,if $a, b, c$ are in

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