The different equation of all straight lines which are at a constant distance from the origin is

Let the straight line be $\mathrm{xcos}\mathrm{\alpha }+\mathrm{ysin}\mathrm{\alpha }=\mathrm{P}$ where $\mathrm{\alpha }$ is arbitrary constant differentiate w.r.t ‘x’ $\cos \mathrm{\alpha }+\sin {\mathrm{\alpha y}}_1=0\dots \dots \dots ..\left(1\right);\tan \mathrm{\alpha }=\frac{-1}{{\mathrm{y}}_1}\dots \dots \dots ..\left(2\right)$

From (1 ) and (2)

   $$\begin{gathered} \mathrm{x}\left(\frac{-{\mathrm{y}}_1}{\sqrt{1+{\mathrm{y}}_1^2}}\right)+\mathrm{y}\left(\frac{1}{\sqrt{1+{\mathrm{y}}_1^2}}\right)=\mathrm{P} \\ \Rightarrow {\left(\mathrm{y}-{\mathrm{xy}}_1\right)}^2={\mathrm{p}}^2\left(1+{\mathrm{y}}_1^2\right) \end{gathered}$$