Question

For what value of $\lambda$ are the three lines $2x-5y+3=0,5x-9y+\lambda =0$ and $x-2y+1=0$ concurrent?

Answer 1

$2x-5y+3=0...\left(i\right)$

$5x-9y+\lambda =0...\left(ii\right)$

$x-2y+1=0...\left(iii\right)$

It is given that the three lines are concurrent

$\therefore$ $\left|\begin{array}{ccc}2& -5& 3\\ 5& -9& \lambda \\ 1& -2& 1\end{array}\right|=0$

$\Rightarrow 2(-9+.2\lambda )+5(5-\lambda )+3(-10+9)=0$

$\Rightarrow -18+4\lambda +25-5\lambda -3=0$

$\Rightarrow \lambda =4$