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Example 16 - Show that two lines a1x + b1y + c1 = 0 - Two lines // or/and prependicular

  1. Chapter 10 Class 11 Straight Lines
  2. Serial order wise
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Example 16 Show that two lines a1x + b1y + c1 = 0 and a2 x + b2 y + c2 = 0 , where b1, b2 โ‰  0 are: (i) Parallel if ๐‘Ž_1/๐‘_1 = ๐‘Ž2/๐‘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 = โˆ’c1 โˆ’ a1 x b1y = โˆ’ a1 x โˆ’ c1 y = ( โˆ’ ๐‘Ž_1 ๐‘ฅ โˆ’ ๐‘_1)/๐‘_1 y = ((โˆ’๐‘Ž_1)/๐‘_1 )x โ€“ (๐‘_1/๐‘_1 ) The above equation is of the form y = mx + c where m is the slope Thus, Slope of line (1) = m1 = (โˆ’๐‘Ž_1)/๐‘_1 From (2) a2x + b2y + c2 = 0 b2y = โˆ’ c2 โˆ’ a2 x b2y = โˆ’ a1 x โˆ’ c2 y = ( โˆ’ ๐‘Ž_2 ๐‘ฅ โˆ’ ๐‘_2)/๐‘_2 y = ((โˆ’๐‘Ž_2)/๐‘_2 )x + (๐‘_2/๐‘_2 ) The above equation is of the form y = mx + c where m is the slope Thus, Slope of line (2) = m2 = (โˆ’๐‘Ž_2)/๐‘_2 Since line (1) & (2) are parallel. So, m1 = m2 (โˆ’๐‘Ž_1)/๐‘_1 = (โˆ’๐‘Ž_2)/๐‘_2 ( ๐‘Ž_1)/๐‘_1 = ๐‘Ž_2/๐‘_2 Hence proved Example 16 Show that two lines a1x + b1y + c1 = 0 and a2 x + b2 y + c2 = 0 , where b1, b2 โ‰  0 are: (ii) Perpendicular if a1a2 + b1b2 = 0 . If two lines are perpendicular, then product of their slope is equal to โˆ’1 Since line (1) & (2) are perpendicular โ‡’ (Slope of line 1) ร— (Slope of line 2) = โˆ’1 m1 ร— m2 = โˆ’ 1 ( โˆ’ ๐‘Ž_1)/๐‘_1 ร— ( โˆ’ ๐‘Ž_2)/๐‘_2 = โˆ’ 1 ( ๐‘Ž_1)/๐‘_1 ร— ๐‘Ž_2/๐‘_2 = โˆ’ 1 a1a2 = โˆ’b1b2 a1a2 + b1b2 = 0 Hence proved

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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