C1:x^2+y^2=25 , C2:x^2+y^2-2x-4y-7=0 be two circles intersecting each ither at A and B. What is the point of intersection of tangents of C1 at A and B?

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$$\begin{gathered} \mathrm{We} \mathrm{have} \\ {\mathrm{C}}_1:{\mathrm{x}}^2+{\mathrm{y}}^2-25=0 \\ {\mathrm{C}}_2:{\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}-4\mathrm{y}-7=0 \\ \mathrm{Equation} \mathrm{of} \mathrm{common} \mathrm{chord} \mathrm{AB} \mathrm{is} \\ {\mathrm{C}}_1-{\mathrm{C}}_2=0 \\ \left({\mathrm{x}}^2+{\mathrm{y}}^2-25\right)-\left({\mathrm{x}}^2+{\mathrm{y}}^2-2\mathrm{x}-4\mathrm{y}-7\right)=0 \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2-25-{\mathrm{x}}^2-{\mathrm{y}}^2+2\mathrm{x}+4\mathrm{y}+7=0 \\ \Rightarrow 2\mathrm{x}+4\mathrm{y}-18=0 \\ \Rightarrow \mathrm{x}+2\mathrm{y}-9=0...\left(\mathrm{i}\right) \\ \mathrm{Now} \mathrm{tangents} \mathrm{are} \mathrm{made} \mathrm{at} \mathrm{A} \mathrm{and} \mathrm{B}. \mathrm{If} \mathrm{tangents} \mathrm{meet} \mathrm{at} \mathrm{P}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right), \mathrm{then} \\ \mathrm{AB} \mathrm{is} \mathrm{chord} \mathrm{of} \mathrm{contact} \mathrm{with} \mathrm{respect} \mathrm{to} \mathrm{P} \\ \mathrm{equation} \mathrm{of} \mathrm{chord} \mathrm{of} \mathrm{contact} \mathrm{is} \\ \mathrm{T}=0 \\ {\mathrm{xx}}_1+{\mathrm{yy}}_1-25=0...\left(\mathrm{ii}\right) \\ \mathrm{Compare} \mathrm{with} \left(\mathrm{i}\right) \mathrm{and} \left(\mathrm{ii}\right) \\ \frac{{\mathrm{x}}_1}{1}=\frac{{\mathrm{y}}_1}{2}=\frac{-25}{-9} \\ {\mathrm{x}}_1=\frac{25}{9} \\ {\mathrm{y}}_1=\frac{50}{9} \\ \left({\mathbf{x}}_{\mathbf{1}}\mathbf{,}{\mathbf{y}}_{\mathbf{1}}\right)\mathbf{=}\left(\frac{\mathbf{25}}{\mathbf{9}}\mathbf{,}\frac{\mathbf{50}}{\mathbf{9}}\right) \end{gathered}$$

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