If the system of linear equations 2 x + 2 a y + a z = 0 2 x + 3 b y + b z = 0 2 x + 4 c y + c z = 0 , where a , b , c ∈ R are non-zero and distinct; has a non-zero solution, then

Question

If the system of linear equations 2 x + 2 a y + a z = 0 2 x + 3 b y + b z = 0 2 x + 4 c y + c z = 0 , where a , b , c ∈ R are non-zero and distinct; has a non-zero solution, then, 1 a , 1 b , 1 c are in A . P ., a , b , c are in G . P ., a + b + c = 0, a , b , c are in A . P .

MARKS App · Accepted Answer

2 2 a a 2 3 b b 2 4 c c = 0 1 2 a a 1 3 b b 1 4 c c = 0 3 b c - 4 b c - 2 a c - 4 a c + 2 a b - 3 a b = 0 - b c + 2 a c - a b = 0 a b + b c = 2 a c a , b , c in H . P . ⇒ 1 a , 1 b , 1 c in A . P .

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