Find the distance between parallel lines
(i) `15 x" "+" "8y" "" "34" "=" "0` and `15 x" "+" "8y" "+" "31" "=" "0 `
(ii) `|(x" "+" "y)" "+" "p" "=" "0|" "(x" "+" "y)" "" "r" "=0` .

Explanation

`(i)` Given lines :
`15x+8y-34=0`……`(1)`
and `15x+8y+31=0`……..`(2)`
Put `x=0` in equation `(1)`,
`0+8y-34=0`
`implies y=(17)/(4)`
`:.` A point on equation `(1)` is , `(0,(17)/(4))`
Length of perpendicular from `(0,(17)/(4))` to second line
`=|(0+8xx(17)/(4)+31)/(sqrt(15^(2)+8^(2)))|=(65)/(17)` unit
`(ii)` Given lines :
`l(x+y)+p=0implieslx+ly+p=0`..........`(1)`
and `l(x+y)-r=0implieslx+ly-r=0`..........`(2)`
Put `y=0` in equation `(2)`
`lx+0-rimpliesx=(r )/(l)`
`:.` A point on equation `(2)` is `((r )/(l),0)`.
length of perpendicular from `((r )/(l),0)` to first line
`=|(l*(r )/(l)+0+p)/(sqrt(l^(2)+l^(2)))|=|(r+p)/(lsqrt(2))|` unit