Question

The value of $\lambda$, for which the three lines $2x-5y+3=0, 5x-9y+\lambda=0$ and $x-2y+1=0$, are concurrent, is

Answer 1

4

Answer 2

Three lines are concurrent if they intersect at a single point. First, find the intersection point of the two lines that do not contain $\lambda$: $2x-5y+3=0$ and $x-2y+1=0$. Solving these equations yields the point $(1,1)$. Substitute this point $(1,1)$ into the third line's equation, $5x-9y+\lambda=0$, to find $\lambda$. This gives: $5(1) - 9(1) + \lambda = 0$ $5 - 9 + \lambda = 0$ $-4 + \lambda = 0$ Thus, $\lambda=4$.