Given:

ax + 2y + 1 = 0

bx + 3y + 1 = 0

cx + 4y + 1 = 0

To prove:

The straight lines ax + 2y + 1 = 0, bx + 3y + 1 = 0 and cx + 4y + 1 = 0 are concurrent.

Concept Used:

If three lines are concurrent then determinant of equation is zero.

Explanation:

The given lines can be written as follows:

ax + 2y + 1 = 0 … (1)

bx + 3y + 1 = 0 … (2)

cx + 4y + 1 = 0 … (3)

Consider the following determinant.

Applying the transformation R₁→R₁-R₂ and R₂→R₂-R₃,

⇒ (-a + b + b – c) = 2b – a – c

Given:
2b = a + c

⇒ a + c –a – c = 0

Hence proved, the given lines are concurrent, provided 2b = a + c.