Question

The determinant $\left|\begin{array}{ccc}a& b& a\alpha +b\\ b& c& b\alpha +c\\ a\alpha +b& b\alpha +c& 0\end{array}\right|$ 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 $a{x}^2+2bx+c$

Answer 1

Answer: (B), (D)

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