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

Misc 13 - Chapter 9 Class 11 Sequences and Series - Part 2
Misc 13 - Chapter 9 Class 11 Sequences and Series - Part 3

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Misc 13 If , (a+bx)/(aβˆ’bx) = (b+cx)/(bβˆ’cx) = (c+dx)/(aβˆ’dx) (x β‰  0)then show that a, b, c and d are in G.P. Introduction Componendo dividendo If π‘₯/𝑦 = π‘Ž/𝑏 Applying componendo dividendo (π‘₯ + 𝑦)/(π‘₯ βˆ’ 𝑦) = (π‘Ž + 𝑏)/(π‘Ž βˆ’ 𝑏) Eg: Taking 1/2 = 4/8 (1+ 2)/(1 βˆ’ 2) = (4 + 8)/(4 βˆ’ 8) 3/(βˆ’1) = 12/(βˆ’4) -3 = -3 Misc 13 If , (a+bx)/(aβˆ’bx) = (b+cx)/(bβˆ’cx) = (c+dx)/(aβˆ’dx) (x β‰  0)then show that a, b, c and d are in G.P. We have (a+bx)/(aβˆ’bx) = (b+cx)/(bβˆ’cx) = (c+dx)/(c βˆ’ dx) & we want to show that a, b, c, d are in G.P. Taking (a+bx)/(aβˆ’bx) = (b+cx)/(bβˆ’cx) = (c+dx)/(c βˆ’ dx) Applying componendo dividendo (a + bx + a βˆ’ bx)/((a + bx) βˆ’(aβˆ’bx)) = (b + cx + (b βˆ’ cx))/(b + cx βˆ’(b βˆ’ cx)) = (c + dx + (c βˆ’ dx))/(c + dx βˆ’ (c βˆ’ dx)) (a + a + bx βˆ’ bx)/(𝑏π‘₯+ bx βˆ’ a + a ) = (b + b + cx βˆ’ cx)/(cx + cx βˆ’ 𝑏 + 𝑏) = (c + dx + c βˆ’ dx)/(dx + dx βˆ’ 𝑐 + 𝑐) (2π‘Ž+0)/(2𝑏π‘₯+0) = (2𝑏 + 0)/(2𝑐π‘₯ + 0) = (2𝑐+0)/(2𝑑π‘₯+0) 2π‘Ž/2𝑏π‘₯ = 2𝑏/2𝑐π‘₯ = 2𝑐/2𝑑π‘₯ π‘Ž/𝑏π‘₯ = 𝑏/𝑐π‘₯ = 𝑐/𝑑π‘₯ 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

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.