Answer :

Given:


ax + a2y + 1 = 0


bx + b2y + 1 = 0


cx + c2y + 1 = 0


To prove:


At least two of three constant a, b, c are equal.


Concept Used:


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


Explanation:


The given lines can be written as follows:


ax + a2y + 1 = 0 … (1)


bx + b2y + 1 = 0 … (2)


cx + c2y + 1 = 0 … (3)


The given lines are concurrent.



Applying the transformation R1R1-R2 and R2R2-R3:




(a – b)(b – c)(c – a) = 0


a – b = 0 or b – c = 0 or c – a = 0


a = b or b = c or c = a


Hence proved, atleast two of the constants a,b,c are equal .


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