Given:

\[2x~-~5y\text{ }+\text{ }3\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 1 \right)\]

\[5x~-~9y\text{ }+~\lambda ~=\text{ }0\text{ }\ldots \text{ }\left( 2 \right)\]

And ,

\[x~-~2y\text{ }+\text{ }1\text{ }=\text{ }0\text{ }\ldots \text{ }\left( 3 \right)\]

It is given that the three lines are concurrent.

Now, consider the following determinant:

\[2\left( -9\text{ }+\text{ }2\lambda  \right)\text{ }+\text{ }5\left( 5\text{ }~\lambda  \right)\text{ }+\text{ }3\left( -10\text{ }+\text{ }9 \right)\text{ }=\text{ }0\]

\[-18\text{ }+\text{ }4\lambda ~+\text{ }25\text{ }\text{ }5\lambda ~\text{ }3\text{ }=\text{ }0\]

\[\lambda ~=\text{ }4\]

∴ The value of λ is 4.