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  4. If the system of linear equations x + 2ay + az = 0, x + 3 by + bz = 0 , x + 4cy + cz = 0 has a non-zero solution, then a, b, c

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Q. If the system of linear equations $ x + 2ay + az = 0, x + 3 by + bz = 0 , x + 4cy + cz = 0$ has a non-zero solution, then a, b, c

AIEEEAIEEE 2003Determinants

Solution:

For non-zero solution $\begin{vmatrix}1&2a&a\\ 1&3b&b\\ 1&4c&c\end{vmatrix} = 0$
$\Rightarrow \, 3bc - 4bc - (2ac - 4ac) + 2ab - 3ab$ = 0
$\Rightarrow \, - bc + 2 ac - ab$ = 0
$\Rightarrow \, \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \, \Rightarrow \, a, b, c$ are in H.P.