[Math] Why does subtracting the equations of two intersecting circles give the line that intersects the points of intersection

Question

Why does subtracting the equations of two intersecting circles give the line that intersects the points of intersection?

In other words, why does the elimination method give a line that also intersects the point of intersection?

Answer 1

Any point that satisfies both equations also satisfies any linear combination of the two equations. Put another way, the curve described by the combined equation also passes through the intersection points of the two circles. If you eliminate the quadratic terms by subtracting one circle equation from the other, you’re left with the equation of a straight line.

Proving the first statement is quite straightforward: the two equations can be put in the form $f(x,y)=0$ and $g(x,y)=0$. If some point $(x_0,y_0)$ satisfies these equations, then $$\lambda f(x_0,y_0)+\mu g(x_0,y_0)=0\lambda+0\mu=0.$$