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 R1→R1-R2 and R2→R2-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 .