Question

If a : b = c : d, show that (a - c) b² : (b - d) cd = (a² - b² - ab) : (c² - d² - cd).

Answer 1

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.