Question

If the system of linear equations
$2x +2ay +az = 0$
$2x +3by + bz = 0$
$2x + 4cy + cz=0$,
where a, b, c $\epsilon$ 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
$\begin{vmatrix}2&2a&a\\ 2&3b&b\\ 2&4c&c\end{vmatrix} = 0, \Rightarrow \begin{vmatrix}1&2a&a\\ 0&3b-2a&b-a\\ 0&4c-2a&c-a\end{vmatrix} = 0$
$\Rightarrow \left(3b - 2a\right) \left(c -a\right) - \left(b - a\right) \left(4c - 2a\right) = 0$
$\Rightarrow 2ac = bc + ab$
$\Rightarrow \frac{2}{b} = \frac{1}{a} +\frac{1}{c}$
Hence $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P.