# If a, b, c are in A. P., prove that the straight lines ax + 2y + 1 = 0, bx + 3y + 1 = 0 and cx + 4y + 1 = 0 are concurrent.

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 R1R1-R2 and R2R2-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.

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