# 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 $\lambda$?

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 $$3x+4y+5+ λ(2x+y+4)=0$$

• Can you link to an example? Jul 3, 2020 at 10:24

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) .