If a ≠ 6, b, c satisfy then abc =

Engineering

Mathematics

Introduction to Determinants

Properties of Determinant

maximum and minimum of quadratic and rational expression

Question

If a ≠ 6, b, c satisfy $∣ \begin{array}{ccc} a & 2 b & 2 c \\ 3 & b & c \\ 4 & a & b \end{array} ∣ = 0$ then abc =

b³

a + b + c

ab + b – c

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Expanding the Determinant we get,

a(b² – ac) – 2b(3b – 4c) + 2c(3a – 4b) = 0

⇒ ab² – a²c – 6b² + 8bc + 6ca – 8bc = 0

⇒ ab² – 6b² = a²c – 6ca

⇒ (a – 6)b² = (a – 6)ca ⇒ b² = ca

Multilpying b both sides we get

b³ = abc