Let the lines 3x – 4y – α = 0, 8x – 11y – 33 = 0, and 2x – 3y + $\mathrm{\lambda }$ = 0 be concurrent. If the image of the point (1, 2) in the line 2x – 3y + $\mathrm{\lambda }$ = 0 is $\left(\frac{57}{13},\frac{-40}{13}\right)\text{, then }\left|\mathrm{\alpha \lambda }\right|$ is equal to :

$\begin{array}{l}\because \mathrm{PM}=\mathrm{QM}\\ \text{ So, }\mathrm{M}\left(\frac{\frac{57}{13}+1}{2},\frac{\frac{-40}{13}+2}{2}\right)\\ =\left(\frac{35}{13},\frac{-7}{13}\right)\end{array}$
M lies on the time
$\begin{array}{l}2\mathrm{x}-3\mathrm{y}+\mathrm{\lambda }=0\\ 2\left(\frac{35}{13}\right)-3\left(\frac{-7}{13}\right)+\mathrm{\lambda }=0\\ \mathrm{\lambda }=-\frac{70}{13}+\frac{21}{13}\\ =\frac{-91}{13}=-7\\ \left|\begin{array}{ccc}3& -4& -\mathrm{\alpha }\\ 8& -11& -33\\ 2& 3& \mathrm{\lambda }\end{array}\right|=0\end{array}$
$\begin{array}{l}\Rightarrow 3(-11\mathrm{\lambda }-99)+4(8\mathrm{\lambda }+66)-\mathrm{\alpha }(-24+22)=0\\ \Rightarrow 33\mathrm{\lambda }-297+32\mathrm{\lambda }+264+24\mathrm{\alpha }-22\mathrm{\alpha }=0\\ \Rightarrow -\mathrm{\lambda }+2\mathrm{\alpha }-33=0\\ \therefore \mathrm{\lambda }=-7\\ -(-7)+2\mathrm{\alpha }-33=0\\ 2\mathrm{\alpha }=26\\ \mathrm{\alpha }=13\\ \therefore \left|\mathrm{\alpha \lambda }\right|=|13\times (-7\left)\right|\\ =91\end{array}$