Find the condition for two lines `a_(1)x+b_(1)y+c_(1)=0` and `a_(2)x+b_(2)y+c_(2)=0` to be
`(i)` parallel
`(ii)` perpendicular

Explanation

Equation of first line
`a_(1)x+b_(1)y+c_(1)=0`
Convert it into slope intercpt form
`b_(1)y=-a_(1)x-c_(1)`
`implies y=-(a_(1))/(b_(1))*x-(c_(1))/(b_(1))`
Slope `m_(1)=-(a_(1))/(b_(1))`
Equation of second line
`a_(2)x+b_(2)y+c_(2)=0`
`implies b_(2)y=-a_(2)x-c_(2)`
`implies y=-(a_(2))/(b_(2))x-(c_(2))/(b_(2))`
and slope `m_(2)=(-a_(2))/(b_(2))`
`(i)` For parallel lines
`m_(1)=m_(2)`
`implies (-a_(1))/(b_(1))=(-a_(2))/(b_(2))`
`impliesa_(1)b_(2)=a_(2)b_(1)` which is the required condition.
`(ii)` For perpendicular lines
`m_(1)*m_(2)=-1`
`implies ((-a_(1))/(b_(1)))((-a_(2))/(b_(2)))=-1`
`implies a_(1)a_(2)=-b_(1)b_(2)`
`implies a_(1)a_(2)+b_(1)b_(2)=0`
which is the required condition.