JEE Main & Advanced Mathematics Circle and System of Circles Image of The Circle by The Line Mirror

Let the circle be \[{{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0\] and line mirror \[lx+my+n=0\]. In this condition, radius of circle remains unchanged but centre changes. Let the centre of image circle be \[({{x}_{1}},\,\,{{y}_{1}})\]. Slope of \[{{C}_{1}}{{C}_{2}}\times \] (slope of \[lx+my+n=0)=-1\]            …..(i)

and mid point of \[{{C}_{1}}(-g,\,-f)\] and \[{{C}_{2}}({{x}_{1}},{{y}_{1}})\] lies on \[lx+my+n=0\]

i.e.,  \[l\,\left( \frac{{{x}_{1}}-g}{2} \right)\,+m\,\left( \frac{{{y}_{1}}-f}{2} \right)\,+\,n=0\]                   …..(ii)

Solving (i) and (ii), we get \[({{x}_{1}},\,\,{{y}_{1}})\]

\[\therefore \] Required image circle is \[{{(x-{{x}_{1}})}^{2}}+{{(y-{{y}_{1}})}^{2}}={{r}^{2}}\],  where \[r=\sqrt{({{g}^{2}}+{{f}^{2}}-c)}\]