Browse Questions

# 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

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

Given :
$\begin{vmatrix}a&b&a\alpha+b\\b&c&b\alpha+c\\a\alpha+b&b\alpha+c&0\end{vmatrix}=0$
Operating $C_3\rightarrow C_3-C_1\alpha-C_2$
We get,
$\begin{vmatrix}a &b&0\\b&c&0\\a\alpha+b&b\alpha+c&-(a\alpha^2+b\alpha+b\alpha+c)\end{vmatrix}=0$
$(ac-b^2)(a\alpha^2+2b\alpha+c)=0$
$\Rightarrow (ac-b^2)=0$ or $a\alpha^2+2b\alpha+c=0$
$\Rightarrow a,b,c$ are in G.P
Hence (b) is the correct answer.