JEE Main & Advanced Mathematics Circle and System of Circles Co-axial System of Circles

A system (or a family) of circles, every pair of which have the same radical axis, are called co-axial circles.

(1) The equation of a system of co-axial circles, when the equation of the radical axis and of one circle of the system are \[P\equiv lx+my+n=0\], \[S\equiv {{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] respectively, is  \[S+\lambda P=0\,\,\] \[(\lambda \] is an arbitrary constant).

(2) The equation of a co-axial system of circles, where the equation of any two circles of the system are

\[{{S}_{1}}\equiv {{x}^{2}}+{{y}^{2}}+2{{g}_{1}}x+2{{f}_{1}}y+{{c}_{1}}=0\]

and  \[{{S}_{2}}\equiv {{x}^{2}}+{{y}^{2}}+2{{g}_{2}}x+2{{f}_{2}}y+{{c}_{2}}=0\]

Respectively, is \[{{S}_{1}}+\lambda \,({{S}_{1}}-{{S}_{2}})=0\]

or \[{{S}_{2}}+{{\lambda }_{1}}\,({{S}_{1}}-{{S}_{2}})=0\]

Other form \[{{S}_{1}}+\lambda {{S}_{2}}=0,\,\,\,\,\,(\lambda \ne -1)\]

(3) The equation of a system of co-axial circles in the simplest form is \[{{x}^{2}}+{{y}^{2}}+2gx+c=0\], where g is a variable and c is a constant.