Question

Let U₁ ≡ a₁x + b₁y + c₁ = 0, U₂ ≡ a₂x + b₂y + c₂ = 0 and U₃ = a₃x + b₃y + c₃ = 0 be three lines such that no two are parallel. If there exists non-zero real numbers λ₁, λ₂ and λ₃ such that λ₁U₁ + λ₂U₂ + λ₃U₃ = 0, then the three lines U₁ = 0, U₂ = 0 and U₃ = 0 are concurrent.

Answer 1

Suppose λ₁U₁ + λ₂U₂ + λ₃U₃ = 0,where λ₁, λ₂ and λ₃ K3 are non-zero real numbers. Therefore

U₃ = (-λ₁/λ₃) + (-λ₂/λ₃)U₂

which is of the form λ₁U₁ + μU₂ = 0. Hence, U₃ = 0 passes through the point of intersection of the lines U₁ = 0 and U₂ = 0. Therefore, the three lines are concurrent.