# chordal

By the entry, the power of the point $(a,\,b)$ with respect to the circle

 $K_{1}(x,\,y):=(x-x_{1})^{2}+(y-y_{1})^{2}-r_{1}^{2}=0$

is equal to  $K_{1}(a,\,b)$  and with respect to the circle

 $K_{2}(x,\,y):=(x-x_{2})^{2}+(y-y_{2})^{2}-r_{2}^{2}=0$

equal to  $K_{2}(a,\,b)$.  Thus the locus of all points $(x,\,y)$ having the same with respect to both circles is characterized by the equation

 $K_{1}(x,\,y)=K_{2}(x,\,y).$

This reduces to the form

 $2(x_{2}-x_{2})x+2(y_{2}-y_{1})y+k=0,$

and hence the locus is a straight line perpendicular to the of the circles.  This locus is called the chordal or the radical axis of the circles.

Title chordal Chordal 2013-03-22 15:07:50 2013-03-22 15:07:50 PrimeFan (13766) PrimeFan (13766) 9 PrimeFan (13766) Result msc 51M99 msc 51N20 radical axis