Question

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

Answer 1

Answer: (B), (D)

The correct options are
B $a,b,c$ are in G.P
D $\alpha$ is a root of $a{x}^2+2bx+c=0$

$\Delta =\left|\begin{array}{ccc}a& b& a\alpha +b\\ b& c& b\alpha +c\\ a\alpha +b& b\alpha +c& 0\end{array}\right|$
Applying $R_3\to R_3-\alpha R_1-R_2$, we get
$\Delta =\left|\begin{array}{ccc}a& b& a\alpha +b\\ b& c& b\alpha +c\\ 0& 0& -(a{\alpha }^2+2b\alpha +c)\end{array}\right|$
Expanding along $R_3$
$\Delta =(b^2-ac)(a{\alpha }^2+2b\alpha +c)$
Now $\Delta =0$
If $b^2-ac=0$ or $a{\alpha }^2+2b\alpha +c=0$
Hence $a,b,c$ are $G.P.$ or $\alpha$ is root of $a{\alpha }^2+2b\alpha +c=0$