Find the equation of a line passing through the intersection of the lines `y=2(x-1)` and `y=3x-5` and which is at a distance of `(7)/(sqrt(2))` units from origin.

Explanation

Equations of given lines are
`y=2(x-1)`
and `y=3x-5`
On solving
Points of intersection `=(3,4)`
Equations of a line passig through the point `(3,4)` is
`y-4=m(x-3)`……….`(1)`
`implies mx-y+4-3m=0`
Perpendicular distance from `(0,0)` to line `(1)` is
`|(0-0+4-3m)/(sqrt(m^(2)+1))|=(7)/(sqrt(2))`
`implies ((4-3m)^(2))/(m^(2)+1)=(49)/(2)`
`implies 49(m^(2)+1)=2(9m^(2)-24m+16)`
`implies 49m^(2)+49=18m^(2)-48m=32`
`implies 31m^(2)+48m+17=0`
`implies(31m+17)(m+1)=0`
`implies m=-(17)/(31)` or `m=-1`
From eq. `(1)`
`(-17)/(31)x-y+4-3((-17)/(31))=0`
or `-x-y+4+3=0`
`implies 17x+31y-124-51=0`
or `x+y-7=0`
`implies 17x+31y=175`
or `x+y-7=0`