A general circle through the intersection points of line $L$ and circle $S_1$ has the form $S_1+\lambda L$. What is the significance of $\lambda$?

Question

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 $3x+4y+5+ λ(2x+y+4)=0$

Answer 1

Let us take up the case of lines first. Let $L_1(x,y)$ and $L_2(x,y)$ be two lines which intersect at $(a,b)\\$. Thus $$L_1(a,b)=0\\L_2(a,b)=0$$Now let $L_3(x,y)$ be another line such that $$L_3(x,y)=L_1(x,y)+\lambda L_2(x,y)$$Now, if we are able to show that $L_3$ passes through $(a,b)$,i.e. the intersection point of $L_1$ and $L_2$, our job will be complete.To do this we put $(a,b)$ in our expression for $L_3$ $$L_3(a,b)=L_1(a,b)+\lambda L_2(a,b)$$ $$\Rightarrow L_3(x,y)=0+\lambda .0$$ $$\Rightarrow L_3(x,y)=0$$So as you can see, for any value of $\lambda$, our line $L_3$ always passes through the intersection of lines $L_1$ and $L_2\\$. You can the same with any two curves(e.g. two circles) .