Question

The determinant $\Delta=\begin{vmatrix}a&b&ax+b\\ b&c&bx+c\\ ax+b&bx+c&0\end{vmatrix}$ 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. $a x^{2}+b x+c=0$

Answer 1

Answer: (B)

We have
$\Delta=\begin{vmatrix}a&b&ax+b\\ b&c&bx+c\\ ax+b&bx+c&0\end{vmatrix}$
$=\begin{vmatrix}a&b&ax+b\\ b&c&bx+c\\ 0&0&-\left(ax^{2}+2bx+c\right)\end{vmatrix}$,
[applying $R_{3} \rightarrow R_{3}-x R_{1}$
$\left.-R_{2}\right]$
$=\left(b^{2}-a c\right)\left(a x^{2}+2 b x+c\right)$
Now, $\Delta=0 \Rightarrow b^{2}=a c$ or $a x^{2}+2 b x+c=0$