Question

If the system of linear equations
$2x+2ay+az=0$
$2x+3by+bz=0$
$2x+4cy+cz=0$,
where a, b, c $ϵ$ R are non-zero and distinct; has a non-zero solution, then :

• A. $a+b+c=0$
• B. a, b, c are in A.P
• C. $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P
• D. a, b, care in G.P

Answer 1

Answer: (C)

For non-zero solution
$\left|\begin{array}{ccc}2& 2a& a\\ 2& 3b& b\\ 2& 4c& c\end{array}\right|=0,\Rightarrow \left|\begin{array}{ccc}1& 2a& a\\ 0& 3b-2a& b-a\\ 0& 4c-2a& c-a\end{array}\right|=0$
$\Rightarrow \left(3b-2a\right)\left(c-a\right)-\left(b-a\right)\left(4c-2a\right)=0$
$\Rightarrow 2ac=bc+ab$
$\Rightarrow \frac{2}{b}=\frac{1}{a}+\frac{1}{c}$
Hence $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P.