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.

$$\begin{gathered} \therefore \left|\begin{array}{ccc}2& -5& 3\\ 5& -9& \lambda \\ 1& -2& 1\end{array}\right|=0 \\ \Rightarrow 2\left(-9+2\lambda \right)+5\left(5-\lambda \right)+3\left(-10+9\right)=0 \\ \Rightarrow -18+4\lambda +25-5\lambda -3=0 \\ \Rightarrow \lambda =4 \end{gathered}$$