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)

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)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.