Question

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

• A. a, b, c are in A.P
• B. a, b, c are in G.P
• C. $\alpha$ is a root of the equation $\alpha x^2+bx+c=0$
• D. $(x-\alpha )$ is a factor of $a{x}^2+2bx+c$

Answer 1

Answer: (B), (D)

The correct options are
B a, b, c are in G.P
D $(x-\alpha )$ is a factor of $a{x}^2+2bx+c$

Given that
$\left|\begin{array}{ccc}a& b& a\alpha +b\\ b& c& b\alpha +c\\ a\alpha +b& b\alpha +c& 0\end{array}\right|=0$
Operating $C_3\to C_3-C_1\alpha -C_2$, we get
$\left|\begin{array}{ccc}a& b& 0\\ b& c& 0\\ a\alpha +b& b\alpha +c& -(a{\alpha }^2+b\alpha +b\alpha +c)\end{array}\right|=0$
or $(ac-b^2)(a{\alpha }^2+2b\alpha +c)=0$
Thus, either a,b,c are in G.P or $(x-\alpha )$ is a factor of $a{x}^2+2bx+c$.