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If the system of linear equations $2 \mathrm{x}+2 \mathrm{ay}+\mathrm{az}=0$ ; $2 x+3 b y+b z=0$ ; $2 \mathrm{x}+4 \mathrm{cy}+\mathrm{cz}=0$ ; where $a, b, c \in R$ are non-zero and distinct; has a non-zero solution, then
$a, b, c$ are in $A.P.$
$a + b + c = 0$
$a, b, c$ are in $G.P.$
$\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in $A.P.$
Solution
For non-zero solution
$\left|\begin{array}{ccc}{2} & {2 a} & {a} \\ {2} & {3 b} & {b} \\ {2} & {4 c} & {c}\end{array}\right|=0, \Rightarrow\left|\begin{array}{ccc}{1} & {2 a} & {a} \\ {0} & {3 b-2 a} & {b-a} \\ {0} & {4 c-2 a} & {c-a}\end{array}\right|=0$
$\Rightarrow(3 b-2 a)(c-a)-(b-a)(4 c-2 a)=0$
$\Rightarrow 2 \mathrm{ac}=\mathrm{bc}+\mathrm{ab}$
$\Rightarrow \frac{2}{b}=\frac{1}{a}+\frac{1}{c}$
Hence $\frac{1}{\mathrm{a}}, \frac{1}{\mathrm{b}}, \frac{1}{\mathrm{c}}$ are in A.P.
