The reflection of a point A(1,8) in the line x - 2 y + 10 = 0 is the point B and the reflection of B in the line x – y + 1 = 0 is the point C then  find the radius of the circle circumscribing triangle ABC?

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$$\begin{gathered} \mathrm{Question}\text{'}s \mathrm{data} \mathrm{seems} \mathrm{incorrect}. \\ \mathrm{A}\left(1,8\right) \\ \mathrm{x}-2\mathrm{y}+10=0 \\ \mathrm{Image} \mathrm{is} \mathrm{given} \mathrm{by} \\ \frac{\mathrm{x}-1}{1}=\frac{\mathrm{y}-8}{-2}=-\frac{2\left(1-2\times 8+10\right)}{1^2+{\left(-2\right)}^2} \\ \Rightarrow \frac{\mathrm{x}-1}{1}=\frac{\mathrm{y}-8}{-2}=-\frac{2\left(-5\right)}{5} \\ \Rightarrow \frac{\mathrm{x}-1}{1}=\frac{\mathrm{y}-8}{-2}=2 \\ \Rightarrow \mathrm{x}=2+1 \mathrm{and} \mathrm{y}=-4+8 \\ \Rightarrow \mathrm{x}=3 \mathrm{and} \mathrm{y}=4 \\ \Rightarrow \mathrm{Image} \mathrm{is} \mathrm{B} \left(3,4\right) \\ \mathrm{Question} \mathrm{says} \mathrm{reflection} \mathrm{of} \mathrm{B} \mathrm{in} \mathrm{x}-\mathrm{y}+1=0 \mathrm{is} \mathrm{C} \mathrm{but} \mathrm{itself} \mathrm{lies} \mathrm{on} \mathrm{x}-\mathrm{y}+1=0 \\ \mathrm{so} \mathrm{there} \mathrm{won}\text{'}\mathrm{t} \mathrm{be} \mathrm{any} \mathrm{image}. \end{gathered}$$

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