Question

The determinant $\left|\begin{array}{ccc}a& b& a\alpha +b\\ b& c& b\alpha +c\\ a\alpha +b& b\alpha +c& 0\end{array}\right|=0$. Which of these statement(s) can be possible?

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

Answer 1

Answer: (B), (D)

$a,b,c$ are in $G.P$.
$(x-\alpha )$ is a factor of $a{x}^2+2bx+c=0$.

Expanding by $C_3$,
$=(a\alpha +b)(b^2\alpha +bc-ac\alpha -bc)-(b\alpha +c)(ab\alpha +ac-ba\alpha -b^2)$
$=(a\alpha +b)(b^2\alpha -ac\alpha )-(b\alpha +c)(ac-b^2)$
$=(b^2-ac)(a{\alpha }^2+\alpha b+b\alpha +c)$
The determinant will be $0$ if $(b^2-ac)=0$, which is a geometric mean,
or,
$(a{\alpha }^2+2b\alpha +c)=0$
i.e., if $(x-\alpha )$ is the factor.