Question

If a, b, c, d are in proportion, prove that `(a + c)^3/(b + d)^3 = (a^2(a - c))/(b^2(b - d))`.

Answer 1

`a/b = c/d` = k

a = bk and c = dk

L.H.S.

= `(a + c)^3/(b + d)^3`

= `(bk + dk)^3/(b + d)^3`

= `[k(b + d)]^3/(b + d)^3`

= `(k^3(b + d)^3)/(b + d)^3`

= k³

R.H.S.

= `(a^2(a - c))/(b^2(b - d))`

= `((bk)^2(bk - dk))/(b^2(b - d))`

= `(b^2k^2 . k(b - d))/(b^2(b - d))`

= `(b^2k^3(b - d))/(b^2(b - d))`

= k³

L.H.S. = R.H.S.