Question

If $\left|\begin{array}{ccc}a& b& a\alpha +b\\ b& c& b\alpha +c\\ a\alpha +b& b\alpha +c& 0\end{array}\right|=0$, then a, b, c are in

• A. AP
• B. GP
• C. HP
• D. none

Answer 1

Answer: (B)

GP

$C_3\to C_3-(C_1\alpha +C_2)$
$\left|\begin{array}{ccc}a& b& 0\\ b& c& 0\\ a\alpha +b& b\alpha +c& -(a{\alpha }^2+2b\alpha +c)\end{array}\right|=0$
$=5(a{\alpha }^2+2\alpha +c)(ac-b^2)=0$
$\Rightarrow b^2=ac$
So $a,b,c$ are in G.P
Ans: B