~~A straight line is parallel to the lines 3x-y-3=0 and 3x-y+5=0 and lies between them. Find its equation if its distances from these lines are in the ratio 3:5.

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have} \\ {\mathrm{L}}_1: 3\mathrm{x}-\mathrm{y}-3=0 \\ {\mathrm{L}}_2: 3\mathrm{x}-\mathrm{y}+5=0 \\ \mathrm{Let} \mathrm{equation} \mathrm{of} \mathrm{third} \mathrm{line} \mathrm{will} \mathrm{be} 3\mathrm{x}-\mathrm{y}=\mathrm{\lambda }...\left(\mathrm{iii}\right) \\ \mathrm{Distance} \mathrm{of} {\mathrm{L}}_1 \mathrm{from} \mathrm{origin}={\mathrm{d}}_1=\frac{\left|-3\right|}{\sqrt{3^2+1}}=\frac{3}{\sqrt{10}} \\ \mathrm{Distance} \mathrm{of} {\mathrm{L}}_2 \mathrm{from} \mathrm{origin}={\mathrm{d}}_2=\frac{\left|5\right|}{\sqrt{3^2+1}}=\frac{5}{\sqrt{10}} \\ \Rightarrow \frac{{\mathrm{d}}_1}{{\mathrm{d}}_2}=\frac{3}{5} \mathrm{which} \mathrm{is} \mathrm{the} \mathrm{ratio} \mathrm{in} \mathrm{which} \mathrm{required} \mathrm{line} \mathrm{must} \mathrm{divide} \mathrm{given} \\ \mathrm{lines}. \\ \mathrm{Hence} \left(\mathrm{iii}\right) \mathrm{must} \mathrm{pass} \mathrm{through} \mathrm{origin}. \\ \Rightarrow 0-0=\mathrm{\lambda }=0 \\ \mathrm{Hence} \\ \mathbf{3}\mathbf{x}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{ }\left(\mathbf{Answer}\right) \end{gathered}$$

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