Find the equation of the line through the point of intersection of, the lines `x-3y + 1 = 0` and `2x + 5y-9-0` and whose distance from the origin is `sqrt5`

Explanation

Let the equation of a line passing through the point of intersection of the lines `x-3y+1=0` and `2x+5y-9=0` be straight line
`(x-3y+1)+lambda(2x+5y-9)=0`
`implies x(2lambda+1)+y(5lambda-3)+(1-9lambda)=0`………`(1)`
Now, the length of perpendicular from origin to this line is
`=|(0+0+(1-9lambda))/(sqrt((2lambda+1)^(2)+(5lambda-3)^(2)))|=sqrt(5)`
`implies (9lambda-1)^(2)-5[2lambda+1)^(2)+(5lambda-3)^(2)]`
`implies 81lambda^(2)-18lambda+1=5(4lambda^(2)+4lambda+1+25lambda^(2)-30lambda+9)`
`implies 81lambda^(2)-18lambda+1=145lambda^(2)-130lambda+50`
`implies 64lambda^(2)-112lambda+49=0`
`implies (8lambda-7)^(2)=0`
`implies lambda=(7)/(8)`
Put `lambda=(7)/(8)` in eq. `(1)`
`x((14)/(8)+1)+y((35)/(8)-3)+(1-(63)/(8))=0`
`implies 22x+11y-55=0`
`implies 2x+y-5=0`