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

Let the slopes of the given lines be `m_(1) and m_(2)` respectively.
Now, `a_(1)x+b_(1)y+c_(1)=0 Rightarrow y=(-a_(1))/(b_(1)).x-(c_(1))/(b_(1))`
`and a_(2)x+b_(2)y+c_(2)=0 Rightarrow y=(-a_(2))/(b_(2)).x-(c_(2))/(b_(2))`
`therefore m_(1)=(-a_(1))/(b_(1)) and m_(2)=(-a_(2))/(b_(2))`
(i) Given lines are parallel, if `m_(1)=m_(2)` which given
`(-a_(1))/(b_(1))=(-a_(2))/(b_(2)) or (a_(1))/(a_(2))=(b_(1))/(b_(2))`
(ii) Given line are perpendicular, `if m_(1)m_(2)=-1` which gives
`((-a_(1))/(b_(1))).((-a_(2))/(b_(2)))=-1 or a_(1)a_(2)+b_(1)b_(2)=0`