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, 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 $a{x}^2+bx+x=0$
• 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 $a{x}^2+2bx+c$.

Operating $R_3-(\alpha R_1+R_2)$ and expanding, we shall easily get $\Delta =-(a{\alpha }^2+2b\alpha +c)(ac-b^2)$.
Hence $\Delta$ is zero if ac$-b^2=0$
or $a{\alpha }^2+2b\alpha +c=0$, i.e., a, b, c
are in G.P. or $x-\alpha$ is a factor of $a{x}^2+2bx+c$