Question

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

Answer 1

Answer: (B)

We have
$\Delta =\left|\begin{array}{ccc}a& b& ax+b\\ b& c& bx+c\\ ax+b& bx+c& 0\end{array}\right|$
$=\left|\begin{array}{ccc}a& b& ax+b\\ b& c& bx+c\\ 0& 0& -\left(a{x}^2+2bx+c\right)\end{array}\right|$,
[applying $R_3\to R_3-x{R}_1$
$-R_2]$
$=\left(b^2-ac\right)\left(a{x}^2+2bx+c\right)$
Now, $\Delta =0\Rightarrow b^2=ac$ or $a{x}^2+2bx+c=0$