If the distance of a point `(x_(1),y_(1))` from each of the two straight lines, which pass through the origin of coordinates, is `delta`, then the two lines are given by

Explanation

Let the line through origin be `y=mxory-mx=0`.
Distance of line `mx-y=0` from `(x_(1),(y_(1))` is
` (mx_(1)-y_(1))/(sqrt(1+m^(2)))=p`(given)
or `m^(2)x_(1)^(2)+y_(1)^(2)-2mx_(1)y_(1)=p^(2)+p^(2)m^(2)`
or `(x_(1)^(2)-p^(2))m^(2)-2mx_(1)y_(1)+y_(1)^(2)-p^(2)=0` (1)
Equation (1) has two roots : `m_(1)andm_(2)`.
Now , the pair of lines is
`(y-m_(1)x)(y-m_(2)x)=0`
or `y^(2)-(m_(1)+m_(2))xy+m_(1)m_(2)x^(2)=0`(2)
Now , from (1),
`m_(1)+m_(2)=(2x_(1)y_(1))/((x_(1)^(2)-p^(2)))`
and `m_(1)m_(2)((y_(1)^(2)-p^(2)))/((x_(1)^(2)-p^(2)))`
Putting these values in (2), we get
`y^(2)-[2x_(1)y_(1)//(x_(1)^(2)-p^(2))]xy+[(y_(1)^(2)-p^(2))//(x_(1)^(2)-p^(2))]x^(2)=0`