The power of a point P with respect to a circle S≡x2+y2+2gx+2fy+c=0 is the value S1=x12+y12+2gx1+2fy1+c obtained by substituting P=(x1,y1) into the left side of the circle's equation; when P lies outside the circle, this power equals the square of the length of the tangent from P to the circle.
The radical axis of two circles S=0 and S′=0 is the locus of points whose power with respect to the two circles is equal -- equivalently, from which the tangent lengths to the two circles (when both are real) are equal. Setting the two powers equal, S=S′, i.e. S−S′=0, and expanding shows that the x2 and y2 terms cancel (both circles have the same leading coefficient 1), leaving a linear equation in x,y:
2(g−g′)x+2(f−f′)y+(c−c′)=0.
So the radical axis is always a genuine straight line, regardless of whether the two circles actually meet in real points; when they do meet, this line is precisely their common chord (or common tangent, if they touch), since every common point automatically has equal (zero) power with respect to both circles in the sense of satisfying both equations.
The radical axis is always perpendicular to the line joining the two centres (−g,−f) and (−g′,−f′): the direction of that line is (g−g′,f−f′) (up to sign), which is exactly the normal direction of the radical axis 2(g−g′)x+2(f−f′)y+(c−c′)=0, whose coefficients of x and y are proportional to (g−g′,f−f′). …