tangents ab and ac are drawn to the circle x^2+y^2-2x+4y+1=0 from a(0,1) then equation of circle passing through a,b and c is

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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ {\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}+4\mathrm{y}+1=0 \\ \mathrm{Circle} \mathrm{pass} \mathrm{through} \mathrm{A}\left(0,1\right) \mathrm{and} \mathrm{chord} \mathrm{of} \mathrm{contact} \mathrm{of} \mathrm{given} \mathrm{circle} \\ \mathrm{Equation} \mathrm{of} \mathrm{chord} \mathrm{of} \mathrm{contact} \mathrm{is} \\ \mathrm{T}=0 \\ {\mathrm{xx}}_1+{\mathrm{yy}}_1-\left(\mathrm{x}+{\mathrm{x}}_1\right)+2\left(\mathrm{y}+{\mathrm{y}}_1\right)+1=0 \\ \left({\mathrm{x}}_1,{\mathrm{y}}_1\right)=\left(0,1\right) \\ 0+\mathrm{y}-\left(\mathrm{x}+0\right)+2\left(\mathrm{y}+1\right)+1=0 \\ \Rightarrow 3\mathrm{y}-\mathrm{x}+3=0 \\ \mathrm{Equation} \mathrm{of} \mathrm{circle} \mathrm{passing} \mathrm{chord} \mathrm{and} \mathrm{circle} \mathrm{is} \\ \mathrm{S}+\mathrm{\lambda L}=0 \\ \Rightarrow \left({\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}+4\mathrm{y}+1\right)+\mathrm{\lambda }\left(3\mathrm{y}-\mathrm{x}+3\right)=0...\left(\mathrm{i}\right) \\ \mathrm{Required} \mathrm{Circle} \mathrm{passes} \mathrm{through} \left(0,1\right) \\ 0+1-0+4+1+\mathrm{\lambda }\left(3+3\right)=0 \\ \Rightarrow \mathrm{\lambda }=-1 \\ \mathrm{Put} \mathrm{in} \left(\mathrm{i}\right) \\ \left({\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}+4\mathrm{y}+1\right)-1\left(3\mathrm{y}-\mathrm{x}+3\right)=0 \\ {\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{-}\mathbf{x}\mathbf{+}\mathbf{y}\mathbf{-}\mathbf{2}\mathbf{=}\mathbf{0} \end{gathered}$$

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