I Family of Circles at Two Points

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As I was flipping through pages of my analytic geometry book from high school, in circle section I stumbled across the formula of "family of circles intersecting at two points" with two circles (##x^2 + y^2 + D_1 x + E_1 y + F_1 = 0## , ##x^2 + y^2 + D_2 x + E_2 y + F_2 = 0##) known to intersect at two points as follows: $$x^2 + y^2 + D_1 x + E_1 y + F_1 + \lambda (x^2 + y^2 + D_2 x + E_2 y + F_2) = 0, \qquad \lambda \setminus \{-1\} \in \mathcal{R}$$ Though at first glance one derivation seems obvious when we just multiply one of those two equations by ##\lambda## and add them up to get the final form, yet this raises a question to me why the final equation involving ##\lambda## has to pass the points that the two circle equations used in its derivation pass.

In light of these, my question is as to a rigorous derivation of this equation in which involves theorems from linear algebra and vector algebra such as linear independence, orthogonality of two vectors by dot product, etc. How can we approach this with respect to what I mentioned above?
 

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As I was flipping through pages of my analytic geometry book from high school, in circle section I stumbled across the formula of "family of circles intersecting at two points" with two circles (##x^2 + y^2 + D_1 x + E_1 y + F_1 = 0## , ##x^2 + y^2 + D_2 x + E_2 y + F_2 = 0##) known to intersect at two points as follows: $$x^2 + y^2 + D_1 x + E_1 y + F_1 + \lambda (x^2 + y^2 + D_2 x + E_2 y + F_2) = 0, \qquad \lambda \setminus \{-1\} \in \mathcal{R}$$ Though at first glance one derivation seems obvious when we just multiply one of those two equations by ##\lambda## and add them up to get the final form, yet this raises a question to me why the final equation involving ##\lambda## has to pass the points that the two circle equations used in its derivation pass.
At the intersection, both coordinates fulfill the equations ##C_1(x,y)=0## and ##C_2(x,y)=0##. Thus ##C_1(x,y)+\lambda C_2(x,y) =0## holds, too. In the other direction, you need to allow variation of ##\lambda## in order to conclude that the two summands have to vanish.
In light of these, my question is as to a rigorous derivation of this equation in which involves theorems from linear algebra and vector algebra such as linear independence, orthogonality of two vectors by dot product, etc. How can we approach this with respect to what I mentioned above?
It is not quite clear to me what "derivation" means here. What is the purpose of the entire thing?
 

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