If , then the family of lines ax + by + c = 0 passes through

Engineering

Mathematics

Introduction to Determinants

Family of Straight lines

domain of function

Question

If $Δ = ∣ \begin{array}{ccc} x + 1 & x + 2 & x + a \\ x + 2 & x + 3 & x + b \\ x + 3 & x + 4 & x + c \end{array} ∣ = 0$ , then the family of lines ax + by + c = 0 passes through

(1, – 2)

(2, – 3)

(0, 0)

(1, – 1)

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$Δ = ∣ \begin{array}{ccc} x + 1 & x + 2 & x + a \\ x + 2 & x + 3 & x + b \\ x + 3 & x + 4 & x + c \end{array} ∣ = 0$

Applying C₂ → C₂ – C₁, C₃ → C₃ – C₁

Δ = ∣ \begin{array}{ccc} x + 1 & 1 & a - 1 \\ x + 2 & 1 & b - 2 \\ x + 3 & 1 & c - 3 \end{array} ∣ = 0

Applying R₂ → R₂ – R₁, R₃ → R₃ – R₁

Δ = ∣ \begin{array}{ccc} x + 1 & 1 & a - 1 \\ 1 & 0 & b - a - 1 \\ 2 & 0 & c - a - 2 \end{array} ∣ = 0

Expanding along C₂
1(c – a – 2) – 2(b – a – 1) = 0

⇒ a – 2b + c = 0
From options for point (1, – 2) lies on the line ax + by + c = 0