Question

For what value of λ are the three lines 2x − 5y + 3 = 0, 5x − 9y + λ = 0 and x − 2y + 1 = 0 concurrent?

Answer 1

Given:
2x − 5y + 3 = 0          ... (1)
5x − 9y + λ = 0          ... (2)
x − 2y + 1 = 0            ... (3)

It is given that the three lines are concurrent

\[\therefore \begin{vmatrix}2 & - 5 & 3 \\ 5 & - 9 & \lambda \\ 1 & - 2 & 1\end{vmatrix} = 0\]

\[ \Rightarrow 2\left( - 9 + 2\lambda \right) + 5\left( 5 - \lambda \right) + 3\left( - 10 + 9 \right) = 0\]

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

\[ \Rightarrow \lambda = 4\]