The straight line passing through the point of intersection of the straight lines

$x-3y+1=0\text{ and }2x+5y-9=0$

and having infinite slope and at a distance 2 units from the origin has the equation 

Any line through the intersection of given lines is 

$\begin{array}{l}P+\lambda Q=0\\ \text{ or }(x-3y+1)+\lambda (2x+5y-9)=0\end{array}$

Its slope will be infinite if it is perpendicular to x-axis i.e. parallel to y-axis. Hence the coefficient of y will be zero

$\therefore -3+5\lambda =0$

or $\lambda =3/5$ Putting for '$\lambda$. the line is $x - 2 = 0$. Clearly its distance from $(0, 0)$is 2.