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

Question

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

MARKS App · Accepted Answer

Case–I: x + 2 y - 3 = 0 & 2 λ x + y - 2 = 0 are parallel ⇒ 2 λ 1 = 1 2 ⇒ λ = 0 . 2 5 Case–II: 3 x - y - 1 = 0 & 2 λ x + y - 2 are parallel ⇒ 2 λ 3 = 1 - 1 ⇒ λ = 1 . 5 Case–III: Lines x + 2 y - 3 = 0 , 3 x - y - 1 = 0 & 2 λ x + y - 2 = 0 are concurrent ⇒ 2 λ 1 - 2 1 2 - 3 3 - 1 - 1 = 0 ⇒ - 10 λ - 8 + 14 = 0 ⇒ λ = 0 . 6 ⇒ 0 . 25 + 1.5 + 0 . 6 = 2.3 5

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