Answer :

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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