If the straight lines ax + amy + 1 = 0,bx + ( | vedclass

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(D) For the lines to be concurrent,the determinant of the coefficients must be zero: $D = \begin{vmatrix} a & am & 1 \ b & b(m+1) & 1 \ c & c(m+2) &

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$H.P.$ for all $m$

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(D) For the lines to be concurrent,the determinant of the coefficients must be zero: $D = \begin{vmatrix} a & am & 1 \ b & b(m+1) & 1 \ c & c(m+2) & 1 \end{vmatrix} = 0$ Applying column operations $C_2 	o C_2 - mC_1$: $\begin{vmatrix} a & 0 & 1 \ b & b & 1 \ c & 2c & 1 \end{vmatrix} = 0$ Expanding along the first row: $a(b - 2c) - 0 + 1(2bc - bc) = 0$ $ab - 2ac + bc = 0$ $b(a + c) = 2ac$ $b = \frac{2ac}{a + c}$ This implies that $a, b, c$ are in $H.P.$ for all $m 
eq 0$.

If the straight lines $ax + amy + 1 = 0$,$bx + (m + 1)by + 1 = 0$,and $cx + (m + 2)cy + 1 = 0$ $(m \neq 0)$ are concurrent,then $a, b, c$ are in:

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