Question

If the determinant $\left|\begin{array}{ccc}a& b& at-b\\ b& c& bt-c\\ 2& 1& 0\end{array}\right|=0$, if $a,b,c$ are in

• A. $A.P.$
• B. $G.P.$
• C. $H.P.$
• D. $k=1/2$

Answer 1

The correct option is B $G.P.$

$\left|\begin{array}{ccc}a& b& at-b\\ b& c& bt-c\\ 2& 1& 0\end{array}\right|=0$
Expanding it along third row,
$\Rightarrow 2\left[b\right(bt-c)-c(at-b\left)\right]-1\left[a\right(bt-c)-b(at-b\left)\right]=0$
$\Rightarrow 2t(b^2-ac)-(b^2-ac)=0$
$\Rightarrow (2t-1)(b^2-ac)=0$
$\Rightarrow t=\frac{1}{2}$ or $b^2=ac$
If $b^2=ac$ then $a,b,c\in$ G.P.