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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. |
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Answer» 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. |
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