If a ≠ b ≠ c and if ax + by + c = 0, bx + cy + a = 0 and cx + ay + b = 0 are concurrent, then find the value of .

Engineering

Mathematics

Introduction to Determinants

Question

If a ≠ b ≠ c and if ax + by + c = 0, bx + cy + a = 0 and cx + ay + b = 0 are concurrent, then find the value of $2^{a^{2} b^{- 1} c^{- 1}} 2^{b^{2} c^{- 1} a^{- 1}} 2^{c^{2} a^{- 1} b^{- 1}}$ .

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Give, ax + by + c = 0

bx + cy + a = 0

cx + ay + b = 0 are concurrent.

i.e., their determinant is equal to zero.

\Rightarrow ∣ \begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array} ∣ = 0

⇒ a(bc – a²) – b(b² – ac) + c(ab – c²) = 0

⇒ abc – a³ – b³ + abc + abc – c³ = 0

⇒ 3abc – (a³ + b³ + c³) = 0

⇒ a³ + b³ + c³ = 3abc ....(1)

We need to find

2^{a^{2} b^{- 1} c^{- 1}} \cdot 2^{b^{2} c^{- 1} a^{- 1}} \cdot 2^{c^{2} a^{- 1} b^{- 1}}

\Rightarrow 2^{(a^{2} b^{- 1} c^{- 1} + b^{2} c^{- 1} a^{- 1} + c^{2} a^{- 1} b^{- 1})}

\Rightarrow 2^{(\frac{a^{2}}{bc} + \frac{b^{2}}{ca} + \frac{c^{2}}{ab})}

= 2^{(\frac{a^{3} + b^{3} + c^{3}}{abc})}

= 2^{(\frac{3 abc}{abc})}

[From (1)]

= 2³ = 8

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