Check sibling questions

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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Transcript

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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.