The equaiton of the lines through the point (2, 3) and making an intercept of length 2 units between the lines `y+2x=3 and y+2x=5` are

Explanation

Let `l` makes an angle `alpha` with the given parallel lines and intercept `AB` is `3` units.

Now, distance between parallel lines
`=(|5-2|)/(sqrt(1^(2)+2^(2)))=(3)/(sqrt(5))`
`:.sinalpha=(1)/(sqrt(5))`, `cosalpha=(2)/(sqrt(5))`
, and `tan alpha=(1)/(2)`
`implies` Equation of straight line passing through `(2,3)` and making an angle `alpha` with `y+2x=5` is
`(y-3)/(x-2)=tan(theta+alpha)`
`implies(y-3)/(x-2)=(tantheta+tanalpha)/(1-tanthetatanalpha)`
and `(y-3)/(x-2)=(tantheta-tanalpha)/(1+tanthetatanalpha)`
`implies(y-3)/(x-2)=-(3)/(4)` and `(y-3)/(x-2)=(1)/(0)`
`implies3x+4y=18` and `x=2`