Let a , b , c ∈ R + and the system of equations ( 1 - a ) x + y + z = 0 , x + ( 1 - b ) y + z = 0 and x + y + ( 1 - c ) z = 0 has infinitely many solutions, the minimum value of a b c is

Question

Let a , b , c ∈ R + and the system of equations ( 1 - a ) x + y + z = 0 , x + ( 1 - b ) y + z = 0 and x + y + ( 1 - c ) z = 0 has infinitely many solutions, the minimum value of a b c is, 3 3, 9, 27, 3

MARKS App · Accepted Answer

We have, 1 - a x + y + z = 0 x + 1 - b y + z = 0 x + y + 1 - c z = 0 has many solutions ∴ 1 - a 1 1 1 1 - b 1 1 1 1 - c = 0 ⇒ a b + a c + b c = a b c ∴ a b + b c + a c 3 ≥ a 2 b 2 c 2 1 / 3 ⇒ a b c 1 / 3 ≥ 3 ∴ Minimum of a b c = 3 3 = 27

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