If the lines x+2ay+a=0,x+3by+b=0,and x+4cy+c= | vedclass

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(C) The condition for the lines $a_1x + b_1y + c_1 = 0$,$a_2x + b_2y + c_2 = 0$,and $a_3x + b_3y + c_3 = 0$ to be concurrent is that the determinant o

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Harmonic Progression

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(C) The condition for the lines $a_1x + b_1y + c_1 = 0$,$a_2x + b_2y + c_2 = 0$,and $a_3x + b_3y + c_3 = 0$ to be concurrent is that the determinant of their coefficients must be zero:$\begin{vmatrix} 1 & 2a & a \ 1 & 3b & b \ 1 & 4c & c \end{vmatrix} = 0$Expanding the determinant along the first column:$1(3bc - 4bc) - 1(2ac - 4ac) + 1(2ab - 3ab) = 0$$-bc + 2ac - ab = 0$$2ac = ab + bc$Dividing both sides by $abc$ (assuming $a, b, c 
eq 0$):$\frac{2ac}{abc} = \frac{ab}{abc} + \frac{bc}{abc}$$\frac{2}{b} = \frac{1}{c} + \frac{1}{a}$This is the condition for $a, b, c$ to be in Harmonic Progression.

If the lines $x+2ay+a=0$,$x+3by+b=0$,and $x+4cy+c=0$ are concurrent,then $a, b, c$ are in

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