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Question
If a : b = c : d, 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) dk.d)`
= `(b^2k (b -d))/(d2k (b -d)) = b^2/d^2`
R.H.S.
= `(a^2 - b^2 - ab)/(c^2 - d^2 - cd)`
= `(b^2k^2 - b^2 - bk·b)/(d^2k^2 - d^2 - dk·d)`
= `(b^2 (k^2 - k - 1))/(d^2 (k^2 - k - 1)`
= `b^2/d^2`
L.H.S. = R.H.S.
Hence proved.
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