Question

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, then

• A. a, b, c are in AP
• B. a, b, c are in GP
• C. a, b, c are in HP
• D. $(x - \alpha)$ is a factor of $ax^2 + 2bx + c$

Answer 1

Answer: (B), (D)

Given, $ \begin{vmatrix}a& b & a\alpha+b\\b& c& ba+c\\a\alpha+b& b\alpha+c& 0\\\end{vmatrix}=0$
Applying $C_3 \rightarrow C_3 - (\alpha C_1 + C_2)$
Given, $\begin{vmatrix}a& b & 0\\b& c& 0\\a\alpha+b& b\alpha+c& -(a\alpha^2+2b\alpha+c)\\\end{vmatrix}=0$
$\Rightarrow -(a\alpha^2+2b\alpha+c)(ac-b^2)=0$
$\Rightarrow a\alpha^2+2b\alpha+c=0 \ or\ b^2=ac$
$\Rightarrow x-\alpha $ is a factor of $ax^2 + 2bx+ c $ or $a, b, c$, are in GP.