Misc 13 - If (a + bx)/(a - bx) = (b + cx)/(b-cx) = (c + dx) - Miscellaneous

  1. Chapter 9 Class 11 Sequences and Series
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Misc 13 If , (a+bx)/(aโˆ’bx) = (b+cx)/(bโˆ’cx) = (c+dx)/(aโˆ’dx) (x โ‰  0)then show that a, b, c and d are in G.P. Introduction Componendo dividendo If ๐‘ฅ/๐‘ฆ = ๐‘Ž/๐‘ Applying componendo dividendo (๐‘ฅ + ๐‘ฆ)/(๐‘ฅ โˆ’ ๐‘ฆ) = (๐‘Ž + ๐‘)/(๐‘Ž โˆ’ ๐‘) Eg: Taking 1/2 = 4/8 (1+ 2)/(1 โˆ’ 2) = (4 + 8)/(4 โˆ’ 8) 3/(โˆ’1) = 12/(โˆ’4) -3 = -3 Misc 13 If , (a+bx)/(aโˆ’bx) = (b+cx)/(bโˆ’cx) = (c+dx)/(aโˆ’dx) (x โ‰  0)then show that a, b, c and d are in G.P. We have (a+bx)/(aโˆ’bx) = (b+cx)/(bโˆ’cx) = (c+dx)/(c โˆ’ dx) & we want to show that a, b, c, d are in G.P. Taking (a+bx)/(aโˆ’bx) = (b+cx)/(bโˆ’cx) = (c+dx)/(c โˆ’ dx) Applying componendo dividendo (a + bx + a โˆ’ bx)/((a + bx) โˆ’(aโˆ’bx)) = (b + cx + (b โˆ’ cx))/(b + cx โˆ’(b โˆ’ cx)) = (c + dx + (c โˆ’ dx))/(c + dx โˆ’ (c โˆ’ dx)) (a + a + bx โˆ’ bx)/(๐‘๐‘ฅ+ bx โˆ’ a + a ) = (b + b + cx โˆ’ cx)/(cx + cx โˆ’ ๐‘ + ๐‘) = (c + dx + c โˆ’ dx)/(dx + dx โˆ’ ๐‘ + ๐‘) (2๐‘Ž+0)/(2๐‘๐‘ฅ+0) = (2๐‘ + 0)/(2๐‘๐‘ฅ + 0) = (2๐‘+0)/(2๐‘‘๐‘ฅ+0) 2๐‘Ž/2๐‘๐‘ฅ = 2๐‘/2๐‘๐‘ฅ = 2๐‘/2๐‘‘๐‘ฅ ๐‘Ž/๐‘๐‘ฅ = ๐‘/๐‘๐‘ฅ = ๐‘/๐‘‘๐‘ฅ a/b " =" b/c = c/d b/a " =" c/b = d/c Thus, a, b, c & d are in GP because their common ratio is same

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.