For a neq b neq c,if the lines x+2ay+a=0,x+3b | vedclass

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(C) The lines $x+2ay+a=0$,$x+3by+b=0$,and $x+4cy+c=0$ are concurrent if the determinant of their coefficients is zero: $\left|\begin{array}{lll}1 & 2a

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Harmonic progression

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(C) The lines $x+2ay+a=0$,$x+3by+b=0$,and $x+4cy+c=0$ are concurrent if the determinant of their coefficients is zero: $\left|\begin{array}{lll}1 & 2a & a \ 1 & 3b & b \ 1 & 4c & c\end{array}ight|=0$ Applying row operations $R_1 ightarrow R_1-R_2$ and $R_2 ightarrow R_2-R_3$: $\left|\begin{array}{ccc}0 & 2a-3b & a-b \ 0 & 3b-4c & b-c \ 1 & 4c & c\end{array}ight|=0$ Expanding along the first column: $(2a-3b)(b-c) - (3b-4c)(a-b) = 0$ $2ab - 2ac - 3b^2 + 3bc - (3ab - 3b^2 - 4ac + 4bc) = 0$ $2ab - 2ac - 3b^2 + 3bc - 3ab + 3b^2 + 4ac - 4bc = 0$ $-ab - bc + 2ac = 0$ $2ac = ab + bc$ Dividing both sides by $abc$: $\frac{2}{b} = \frac{1}{c} + \frac{1}{a}$ This is the condition for $a, b, c$ to be in Harmonic Progression ($H$.$P$.).

For $a \neq b \neq c$,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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