The distance from a point $\left(\alpha ,\beta \right)$ to a pair of lines passing through the origin is d. Then the equation to the pair of lines is

Let  y=mx be the one of the line

$\begin{array}{l}\therefore m=\raisebox{1ex}{$y$}\!\left/ \!\raisebox{-1ex}{$x$}\right.\\ Given\perp rdis\tan cefrom\left(\alpha ,\beta \right)tomx-y=0isd\\ \Rightarrow \frac{\left|m\alpha -\beta \right|}{\sqrt{m^2-1}}=d\\ \Rightarrow {\left(m\alpha -\beta \right)}^2=d^2\left(m^2+1\right)\\ \Rightarrow {\left(\frac{y}{x}\alpha -\beta \right)}^2=d^2\left(\frac{y^2}{x^2}+1\right)\\ \Rightarrow \frac{\left(y\alpha -x\beta \right)}{x^2}=d^2\left(\frac{y^2+x^2}{x^2}\right)\\ \Rightarrow {\left(\beta x-\alpha y\right)}^2=d^2\left(x^2+y^2\right)\end{array}$