We write a general line $L$ passing through intersection of two lines $L_1$ and $L_2$ as $L= L_1 + (\lambda) L_2$ where $\lambda$ is a variable.
Even in family of circles we write a general circle $S$ passing through points of intersection of a circle $S_1$ and a line $L$ as $S= S_1+(\lambda)L$. But why do we write in this way? What is the significance of $\lambda$ and this form?
For example if we want to find lines through the point of intersection of $3x+4y+5=0$ and $2x+y+4=0$ . The required lines would be obtained by substituting different values of $λ$ in $3x+4y+5+ λ(2x+y+4)=0$