Question

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

Answer 1

`a/b = c/d` = k

a = bk and c = dk

L.H.S.

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

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

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

= `(b^2(k^2 - k + 1))/(d^2(k^2 - k + 1))`

= `b^2/d^2`

R.H.S.

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

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

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

= `(b^2(k^2 - 1))/(d^2(k^2 - 1))`

= `b^2/d^2`

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