# If the three lines ax + a2y + 1 = 0, bx + b2y + 1 = 0 and cx + c2y + 1 = 0 are concurrent, show that at least two of three constant a, b, c are equal.

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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