JEE Main & Advanced Mathematics Circle and System of Circles Family of Circles

(1) The equation of the family of circles passing through the point of intersection of two given circles S = 0 and S' = 0 is given as \[S+\lambda S'=0\],    (where \[\lambda \] is a parameter, \[\lambda \ne -1)\]

(2) The equation of the family of circles passing through the point of intersection of circle S = 0 and a line L = 0 is given as  \[S+\lambda L=0\],   (where \[\lambda \] is a parameter)  

(3) The equation of the family of circles touching the circle  \[S=0\] and the line \[L=0\] at their point of contact P is \[S+\lambda L=0\], (where \[\lambda \] is a parameter)

(4) The equation of a family of circles passing through two given points \[P\,({{x}_{1}},\,{{y}_{1}})\] and \[Q\,({{x}_{2}},\,{{y}_{2}})\] can be written in the form \[(x-{{x}_{1}})\,(x-{{x}_{2}})+(y-{{y}_{1}})\,(y-{{y}_{2}})+\lambda \,\left| \,\begin{matrix} x & y & 1  \\ {{x}_{1}} & {{y}_{1}} & 1  \\ {{x}_{2}} & {{y}_{2}} & 1  \\ \end{matrix}\, \right|\,=0\] , (where \[\lambda \] is a parameter)  

(5) The equation of family of circles, which touch \[y-{{y}_{1}}=m\,(x-{{x}_{1}})\] at \[({{x}_{1}},\,{{y}_{1}})\] for any finite m is \[{{(x-{{x}_{1}})}^{2}}+{{(y-{{y}_{1}})}^{2}}+\lambda \,\{(y-{{y}_{1}})\]\[-m\,(x-{{x}_{1}})\}=0\]

And if \[m\] is infinite, the family of circles is

\[{{(x-{{x}_{1}})}^{2}}+{{(y-{{y}_{1}})}^{2}}+\lambda \,(x-{{x}_{1}})=0\], (where \[\lambda \] is a parameter)  

(6) Equation of the circles given in diagram is  

\[(x-{{x}_{1}})\,(x-{{x}_{2}})+\]\[(y-{{y}_{1}})\,(y-{{y}_{2}})\,\pm \cot \theta \,\{(x-{{x}_{1}})\,(y-{{y}_{2}})\]\[-(x-{{x}_{2}})\,(y-{{y}_{1}})\}=0\]