Question

If `a/b = c/d` then prove that `(6a + 7b)/(6c + 7d) = sqrt((5a^2 + 3b^2)/(5c^2 + 3d^2))`

Answer 1

`a/b = c/d = k`

a = bk

c = dk

L.H.S.

= `(6a + 7b)/(6c + 7d)`

= `(6(bk) + 7b)/(6(dk) + 7d)`

= `(b(6k + 7))/(d(6k + 7))`

= `b/d`

R.H.S.

= `sqrt((5a^2 + 3b^2)/(5c^2 + 3d^2))`

= `sqrt((5(bk)^2 + 3b^2)/(5(dk)^2 + 3d^2))`

= `sqrt((5b^2k^2 + 3b^2)/(5d^2k^2 + 3d^2))`

= `sqrt((b^2(5k^2 + 3))/(d^2(5k^2 + 3)))`

= `sqrt(b^2/d^2)`

= `b/d`

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