Consider the equation of two circles $S_1 = x^2 + y^2 = 4 $ and $ S_2 = x^2 + 2x + 1 + y^2 = 4$,then the equation of common chord is given as:
$$S_1 - S_2=0$$$
And a similar notion can be said for the circle passing through intersection of two circles, say we have two circles $S_1$ and $S_2$ then the family of circles through their intersection is given as:
$$S_1 + \lambda S_2 = 0$$
And also, if we have a line passing through intersection of two lines $(L_1, L_2)$ it's equation is given as:
$$ L_1 + \lambda L_2 = 0$$
So, in all of the above equation's if you look it in the reverse way, you are breaking down curve into a linear combination of two other component curves... which is similar to the notions in linear algebra. So, I wish to ask, is it possible to break down algebraic curves into a linear combination of some basic curves like does their exist 'some basis curves' similar to the basis vectors ?