find the equation of the line passing through the point of intersection of two lines x-3y+1=0 2x+5y-9=0 and whose distance from the origin is 5^{1/2 .}

the general equation of the line which passes through the intersection point of the given lines 
x-3y+1=0 and 2x+5y-9=0 is given by: 
$$\begin{gathered} x-3y+1+\lambda (2x+5y-9)=0 \\ (1+2\lambda )x+(-3+5\lambda )y+(1-9\lambda )=0.........\left(1\right) \end{gathered}$$
the distance of the eq(1) from the origin is $\sqrt{5} unit$.
therefore  $\left|\frac{1-9\lambda }{\sqrt{(1+2\lambda {)}^2+(-3+5\lambda {)}^2}}\right|=\sqrt{5}$
$$\begin{gathered} (1-9\lambda {)}^2=5[(1+2\lambda {)}^2+(-3+5\lambda {)}^2] \\ 1+81{\lambda }^2-18\lambda =5*[1+4{\lambda }^2+4\lambda +9+25{\lambda }^2-30\lambda ] \\ 81{\lambda }^2-18\lambda +1=5*[29{\lambda }^2-26\lambda +10] \\ 81{\lambda }^2-18\lambda +1=145{\lambda }^2-130\lambda +50 \\ 64{\lambda }^2-112\lambda +49=0 \\ (8\lambda {)}^2-2*8\lambda *7+7^2=0 \\ (8\lambda -7{)}^2=0 \\ 8\lambda -7=0\Rightarrow \lambda =\frac{7}{8} \end{gathered}$$
substitute the value of $\lambda$ in eq(1):
$$\begin{gathered} (1+\frac{7}{4})x+(-3+\frac{35}{8})y+(1-\frac{63}{8})=0 \\ \frac{11}{4}x+\frac{11}{8}y-\frac{55}{8}=0 \\ 2x+y-5=0 \end{gathered}$$
which is the required equation of the line.

hope this helps you.