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Question
Given four quantities a, b, c and d are in proportion. Show that: (a – c)b2 : (b – d)cd = (a2 – b2 – ab) : (c2 – d2 – cd)
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Solution
Let `a/b = c/d = k`
`=>` a = bk and c = dk
L.H.S = `((a - c)b^2)/((b -d)cd)`
= `((bk - dk)b^2)/((b - d)d^2k)`
= `b^2/d^2`
R.H.S = `(a^2 - b^2 - ab)/(c^2 - d^2 - cd)`
= `(b^2k^2 - b^2 - bkb)/(d^2k^2 - d^2 - dkd)`
= `(b^2(k^2 - 1 - k))/(d^2(k^2 - 1 - k))`
= `b^2/d^2`
`=>` L.H.S = R.H.S
Hence proved.
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