The straight lines 2x+3y-1=0, 3x+4y-6=0 and ax+by+1=0 are concurrent if the straight line 41x+9y+1=0 passes through the point

Given lines

$\begin{array}{l}2x+3y-1=0\\ 3x+4y-6=0\\ ax+by+1=0\end{array}$

The lines are concurrent iff

$\left|\begin{array}{ccc}2& 3& -1\\ 3& 4& -6\\ a& b& 1\end{array}\right|=0$

$\begin{array}{l}\Rightarrow 2\left(4+6b\right)-3\left(3+6a\right)-1\left(3b-4a\right)=0\\ \Rightarrow 8+12b-9-18a-3b+4a=0\\ \Rightarrow -14a+9b-1=0\\ \Rightarrow 14a-9b+1=0\end{array}$

Comparing with 14x+9y+1=0

       $x=a, y=-b$

$\therefore$The point is (a,-b)