The determinant $\begin{vmatrix}a&b&a\alpha+b\\b&c&b\alpha+c\\a\alpha+b&b\alpha+c&0\end{vmatrix}$ is equal to zero if

Question

$\begin{array}{1 1}(a)\;a,b,c\;are\;in\; A.P\\(b)\;a,b,c\;are\;in \;G.P\\(c)\;a,b,c\;are\;H.P\\(d)\;\alpha\;is\;a\;root\;of\;the\;equation\;ax^2+bx+c=0\end{array}$