If the three lines $a x + a^{2} y + 1 = 0, b x + b^{2} y + 1 = 0$ and $c x + c^{2} y + 1 = 0$ are concurrent, show that atleast two of the three constants a,b,c are equal.

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Solution

Since the lines are concurrent, the equation of the lines taken two at a time should be equal at a point where they meet.
Thus,
$a x + a^{2} y + 1 = b x + b^{2} y + 1 x (a - b) + y (a^{2} - b^{2}) = 0 x (a - b) + y (a - b) (a + b) = 0 (a - b) [x + y (a + b)] = 0 a - b = 0 a = b$
Similarly, by equating 2nd and 3rd equation and 1st and 3rd equation
we get
$b = c$ and $a = c$

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