Example 16 - Chapter 10 Class 11 Straight Lines

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Example 16 Show that two lines a1x + b1y + c1 = 0 and a2 x + b2 y + c2 = 0 , where b1, b2 0 are: (i) Parallel if _1/ _1 = 2/ 2 The given lines are a1x + b1y + c1 = 0 & a2 x + b2 y + c2 = 0 Let slope of line (1) be m1 & slope of line (2) be m2 If two lines are parallel, then their slopes are equal If line (1) & (2) are parallel , then m1 = m2 Finding m1 & m2 From (1) a1x + b1y + c1 = 0 b1y = c1 a1 x b1y = a1 x c1 y = ( _1 _1)/ _1 y = (( _1)/ _1 )x ( _1/ _1 ) The above equation is of the form y = mx + c where m is the slope Thus, Slope of line (1) = m1 = ( _1)/ _1 From (2) a2x + b2y + c2 = 0 b2y = c2 a2 x b2y = a1 x c2 y = ( _2 _2)/ _2 y = (( _2)/ _2 )x + ( _2/ _2 ) The above equation is of the form y = mx + c where m is the slope Thus, Slope of line (2) = m2 = ( _2)/ _2 Since line (1) & (2) are parallel. So, m1 = m2 ( _1)/ _1 = ( _2)/ _2 ( _1)/ _1 = _2/ _2 Hence proved Example 16 Show that two lines a1x + b1y + c1 = 0 and a2 x + b2 y + c2 = 0 , where b1, b2 0 are: (ii) Perpendicular if a1a2 + b1b2 = 0 . If two lines are perpendicular, then product of their slope is equal to 1 Since line (1) & (2) are perpendicular (Slope of line 1) (Slope of line 2) = 1 m1 m2 = 1 ( _1)/ _1 ( _2)/ _2 = 1 ( _1)/ _1 _2/ _2 = 1 a1a2 = b1b2 a1a2 + b1b2 = 0 Hence proved