no of circles touching all the lines x-2y+1=0,2x+y+3=0 and 4x-8y+3=0 is

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$$\begin{gathered} \mathrm{Slope}=\mathrm{m}=-\frac{\mathrm{Coefficient} \mathrm{of} \mathrm{x}}{\mathrm{Coeffient} \mathrm{of} \mathrm{y}} \\ {\mathrm{L}}_1: \mathrm{x}-2\mathrm{y}+1=0 \\ {\mathrm{m}}_1=-\frac{1}{\left(-2\right)}=\frac{1}{2} \\ {\mathrm{L}}_2: 2\mathrm{x}+\mathrm{y}+3=0 \\ {\mathrm{m}}_2=-\frac{2}{1}=-2 \\ {\mathrm{L}}_3: 4\mathrm{x}-8\mathrm{y}+3=0 \\ {\mathrm{m}}_3=-\frac{4}{\left(-8\right)}=\frac{1}{2} \\ \mathrm{As} {\mathrm{m}}_1={\mathrm{m}}_3 \\ \mathrm{Hence} {\mathrm{L}}_1 \mathrm{and} {\mathrm{L}}_3 \mathrm{are} \mathrm{parallel}. \\ \mathrm{As} {\mathrm{m}}_1\times {\mathrm{m}}_2={\mathrm{m}}_3\times {\mathrm{m}}_2=-1 \\ \mathrm{Hence} {\mathrm{m}}_1 \mathrm{and} {\mathrm{m}}_3 \mathrm{will} \mathrm{be} \mathrm{perpendicular} \mathrm{to} {\mathrm{m}}_2. \\ \mathrm{Hence} \mathrm{arrangement} \mathrm{of} \mathrm{lines} \mathrm{will} \mathrm{be} \mathrm{like} \mathrm{given} \mathrm{below}. \end{gathered}$$

$$\begin{gathered} \mathrm{Hence} \mathrm{there} \mathrm{will} \mathrm{be} \mathrm{two} \mathrm{common} \mathrm{circles} \mathrm{touching} \mathrm{all} 3 \mathrm{lines}. \\ \mathrm{Cirlces} \mathrm{will} \mathrm{be} \mathrm{enclosed} \mathrm{between} \mathrm{two} \mathrm{parallel} \mathrm{lines}. \end{gathered}$$

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