दर्शाइए कि दो रेखाएँ`a_(1)x+b_(1)y+c_(1)=0` और `a_(2)x+b_(2)y+c_(2)=0` जहाँ `b_(1),b_(2)ne0`
(i ) समान्तर है यदि `(a_(1))/(b_(1))=(a_(2))/(b_(2))` और (ii ) लम्ब है यदि `a_(1)a_(2)+b_(1)b_(2)=0.`

Explanation

`:' a_(1)x+b_(1)y+c_(1)=0`
`b_(1)x+b_(1)y=-a_(1)x-C_(1)`
`y=(a_(1))/(b_(1))x-(c_(1))/(b_(1))`
इसकी तुलना `y = m _(1 )x +c ` से करने पर ,
`m_(1)=-(a_(1))/(b_(1))`
और `a_(2)x+b_(2)y+c_(2)=0`
`b_(2)y=-a_(2)x-c_(1)`
`y=-(a_(2))/(b_(2))x-(c_(2))/(b_(2))`
पुन : इसकी तुलना `y =m _(2 ) x +c ` से करने पर ,
`m_(2)=-(a_(2))/(b_(2))`
(i ) जब रेखाएँ हो , तब
`m_(1)=m_(2)`
`-(a_(1))/(b_(1))=-(a_(2))/(b_(2))`
या `(a_(1))/(b_(1))=(a_(2))/(b_(2)).`इति सिद्धम (ii ) जब रेखाएँ लंबवत हो , तब `m_(1).m_(2)=-1`
`(-(a_(1))/(b_(1)))(-(a_(2))/(b_(2))=-1`
`(a_(1)a_(2))/(b_(1)b_(2)=-b_(1)b_(2)`
`a_(1)a_(2)+b_(1)b_(2)=0.`इति सिद्धम