The number of real values of λ for which the lines x – 2y + 3 = 0, λx + 3y + 1 = 0 and 4x – λy + 2 = 0 are concurrent is
The given lines are
x − 2y + 3 = 0 … (1)
λx + 3y + 1 = 0 … (2)
4x − λy + 2 = 0 … (3)
It is given that (1), (2) and (3) are concurrent.
∴
⇒ (6 + λ) + 2(2λ – 4) + 3(-λ2 – 12) = 0
⇒ 6 + λ + 4λ – 8 – 3λ2 – 36 = 0
⇒ 5λ – 3λ2 – 38 = 0
⇒ 3λ2 – 5λ + 38 = 0
The discriminant of this equation is 25-4 × 3 × 38 = -431
Hence, there is no real value of λ for which the lines x − 2y + 3 = 0, λx + 3y + 1 = 0 and 4x − λy + 2 = 0 are concurrent.