Question

Find the distance between the pair of parallel lines, y = x-11 & y = x-7

Answer 1

`Given \ the \ equations \ of \ two \ non-vertical \ parallel \ lines:\\\\y = mx+b_1\\y = mx+b_2\\\\the \ distance \ between \ them \ can \ be \ expressed \ as : \\\\d= (|b_(1)-b_(2)|)/( √( m^2+1) )`


`y = x-11 \\ y = x-7 \\\\\\d= (| -11- (-7)|)/( √( 1^2+1) ) =(| -11+7|)/( √( 1+1) ) = (|-4|)/( √(2) ) = (4)/( √(2) )\cdot (√(2))/(√(2))=(4√(2))/(2)=2√(2)`

Answer 2

`k:\ y = x-11\ \ \ \Leftrightarrow\ \ \ x-y-11=0\\ and\\ l:\ y = x-7\ \ \ \Leftrightarrow\ \ \ x-y-7=0\\\\the\ distance:\\\\ d(k;l)= \frac{\big-11-(-7)}{\big{ √(1^2+1^2) }} =\frac{\big}{\big{ √(2) }} =\frac{\big-4}{\big{ √(2) }} =\frac{\big{4\cdot √(2) }}{\big{ √(2)\cdot √(2) }} =\frac{\big{4 √(2) }}{\big{2 }} =2 √(2)`