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

Answer 1

Answer: (B), (E)

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

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

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.