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

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(C) The given lines are concurrent if and only if the determinant of their coefficients is zero:$\begin{vmatrix} 1 & 2a & a \ 1 & 3b & b \ 1 & 4c &

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$H.P.$

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(C) The given lines are concurrent if and only if the determinant of their coefficients is 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$$2ac = b(a + c)$$b = \frac{2ac}{a + c}$This is the condition for $a, b,$ and $c$ to be in $H.P.$ (Harmonic Progression).

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

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