Question

If $\left|\begin{array}{ccc}1& 1& 1\\ 4a& 3b& 2c\\ a& b& c\end{array}\right|=0$, then $a,b,c$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. Not in any progression

Answer 1

The correct option is C H.P.

$\left|\begin{array}{ccc}1& 1& 1\\ 4a& 3b& 2c\\ a& b& c\end{array}\right|=0C_1\to C_1-C_2,C_2\to C_2-C_3\left|\begin{array}{ccc}0& 0& 1\\ 4a-3b& 3b-2c& 2c\\ a-b& b-c& c\end{array}\right|=0\Rightarrow (4ab-4ac-3b^2+3bc)-(3ab-3b^2-2ac+2bc)=0\Rightarrow ab-2ac+bc=0\Rightarrow bc+ab=2ac\Rightarrow \frac{1}{a}+\frac{1}{c}=\frac{2}{b}\Rightarrow a,b,c\text{are in H.P.}$