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The product of all the values of λ , such that the lines x + 2 y - 3 = 0 , 3 x - y - 1 = 0 and λ x + y - 2 = 0 cannot form a triangle, is equal to

Correct answer

1.8

Step-by-step solution

Case I: x + 2 y - 3 = 0 and λ x + y - 2 = 0 are parallel ⇒ λ 1 = 1 2 ⇒ λ = 0 . 5 Case II: 3 x - y - 1 = 0 and λ x + y - 2 = 0 are parallel ⇒ λ 3 = 1 - 1 ⇒ λ = 3 Case III: Lines x + 2 y - 3 = 0 , 3 x - y - 1 = 0 and 2 λ x + y - 2 = 0 are concurrent ⇒ λ 1 - 2 1 2 - 3 3 - 1 - 1 = 0 ⇒ - 5 λ - 8 + 14 = 0 ⇒ λ = 1.2 Hence, the product of all the values of λ = 0 . 5 × 3 × 1.2 = 1.8

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