If the system of homogeneous linear equations $ x+3 b y+b z=0, x+2 a y+a z=0 $, and $ x+4 c y+c z=0 $ has a non-zero solution, then $ a, b, c $

Question

If the system of homogeneous linear equations $ x+3 b y+b z=0, x+2 a y+a z=0 $, and $ x+4 c y+c z=0 $ has a non-zero solution, then $ a, b, c $, Are in A.P., Are in G.P., Are in H.P., Satisfy $ a+2 b+3 c=0 $

MARKS App · Accepted Answer

The system of linear equations has a non-zero solution, then 1 2 a a 1 3 b b 1 4 c c = 0 Applying R 2 → R 2 - R 1 , R 3 → R 3 - R 1 , 1 2 a a 0 3 b - 2 a b - a 0 4 c - 2 a c - a = 0 ⇒ ( 3 b - 2 a ) ( c - a ) - ( 4 c - 2 a ) ( b - a ) = 0 ⇒ 3 b c - 3 b a - 2 a c + 2 a 2 = 4 b c - 2 a b - 4 a c + 2 a 2 ⇒ 4 a c - 2 a c = 4 b c - 2 a b - 3 b c + 3 a b ⇒ 2 a c = b c + a b On dividing by a b c , we get 2 b = 1 a + 1 c Hence, a , b , c are in H . P .

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