Show that the lines `a_(1)x+b_(1)y+c_(1)=0 and a_(2)x+b_(2)y+c_(2)=0,"where" b_(1),b_(2) ne 0 "are (i) parallel, if"(a_(1))/(b_(1))=(a_(2))/(b_(2))" (ii) perpendicular, if "a_(1)a_(2)+b_(1)b_(2)=0`

Explanation

Given lines can be written as
`y=(a_(1))/(b_(1))xx-(c_(1))/(b_(1))" "...(i)`
`andy=-(a_(2))/(b_(2))xx-(c_(2))/(b_(2))" "...(ii)`
Slopes of the lines (i) and (ii) are `m_(1)=-(a_(1))/(b_(1))and m_(2)=-(a_(2))/(b_(2),` respjectively. Now
(a) Lines are parallel, if `m_(1)=m_(2)` which gives
`-(a_(1))/(b_(1))=-(a_(2))/(b_(2))or (a_(1))/(b_(1))=(a_(2))/(b_(2))`
(b) Lines are perpendicular, if `m_(1)m_(2)=-1,` which gives
`(a_(1))/(b_(1)).(a_(2))/(b_(2))=-1or a_(1)a_(2)+b_(1)b_(2)=0`