JEE Main & Advanced Mathematics Circle and System of Circles Radical Centre

The radical axes of three circles, taken in pairs, meet in a point, which is called their radical centre. Let the three circles be 

 \[{{S}_{1}}=0\]   .....(i), \[{{S}_{2}}=0\] .....(ii) and \[{{S}_{3}}=0\]   .….(iii)

Let the straight lines i.e., OL and OM meet in O. The equation of any straight line passing through O is  \[{{x}^{2}}+{{y}^{2}}=\frac{{{a}^{2}}}{2}\], where \[\lambda \] is any constant.

For  \[\lambda =1\], this equation become \[{{S}_{2}}-{{S}_{3}}=0\], which is, equation of ON.

Thus the third radical axis also passes through the point where the straight lines OL and OM meet.

In the above figure O is the radical centre.

Properties of radical centre

(i) Co-ordinates of radical centre can be found by solving the equations \[{{S}_{1}}={{S}_{2}}={{S}_{3}}\].

(ii) The radical centre of three circles described on the sides of a triangle as diameters is the orthocentre of the triangle.