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Example 16 - Show that two lines a1x + b1y + c1 = 0 - Examples

Example 16 - Chapter 10 Class 11 Straight Lines - Part 2
Example 16 - Chapter 10 Class 11 Straight Lines - Part 3
Example 16 - Chapter 10 Class 11 Straight Lines - Part 4

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Example 16 - Chapter 10 Class 11 Straight Lines - Part 5

Example 16 - Chapter 10 Class 11 Straight Lines - Part 6

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Example 16 Show that two lines a1x + b1y + c1 = 0 and a2 x + b2 y + c2 = 0 , where b1, b2 β‰  0 are: (i) Parallel if π‘Ž_1/𝑏_1 = π‘Ž2/𝑏2 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 = βˆ’c1 βˆ’ a1 x b1y = βˆ’a1 x βˆ’c1 y = ( βˆ’π‘Ž_1 π‘₯ βˆ’ 𝑐_1)/𝑏_1 y = ((βˆ’π‘Ž_1)/𝑏_1 ) x –(𝑐_1/𝑏_1 ) The above equation is of the form y = mx + c where m is the slope Thus, Slope of line (1) = m1 = (βˆ’π‘Ž_1)/𝑏_1 From (2) a2x + b2y + c2 = 0 b2y = βˆ’c2 βˆ’ a2 x b2y = βˆ’a2 x βˆ’c2 y = ( βˆ’π‘Ž_2 π‘₯ βˆ’ 𝑐_2)/𝑏_2 y = ((βˆ’π‘Ž_2)/𝑏_2 )x + (𝑐_2/𝑏_2 ) The above equation is of the form y = mx + c where m is the slope Thus, Slope of line (2) = m2 = (βˆ’π‘Ž_2)/𝑏_2 Since line (1) & (2) are parallel. So, m1 = m2 (βˆ’π‘Ž_1)/𝑏_1 = (βˆ’π‘Ž_2)/𝑏_2 ( 𝒂_𝟏)/𝒃_𝟏 = 𝒂_𝟐/𝒃_𝟐 Hence proved Example 16 Show that two lines a1x + b1y + c1 = 0 and a2 x + b2 y + c2 = 0 , where b1, b2 β‰  0 are: (ii) Perpendicular if a1a2 + b1b2 = 0 . If two lines are perpendicular, then product of their slope is equal to βˆ’1 Since line (1) & (2) are perpendicular β‡’ (Slope of line 1) Γ— (Slope of line 2) = βˆ’1 m1 Γ— m2 = βˆ’ 1 ( βˆ’π‘Ž_1)/𝑏_1 Γ— ( βˆ’π‘Ž_2)/𝑏_2 = βˆ’1 ( π‘Ž_1)/𝑏_1 Γ— π‘Ž_2/𝑏_2 = βˆ’1 a1a2 = βˆ’b1b2 a1a2 + b1b2 = 0 Hence proved

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.