Question

If `a/b = c/d = r/f`, prove that `((a^2b^2 + c^2d^2 + e^2f^2)/(ab^3 + cd^3 + ef^3))^(3/2) = sqrt((ace)/(bdf)`

Answer 1

Let `a/b = c/d = r/f = k`
∴ a = bk, c = dk, e = fk
L.H.S.
= `((a^2b^2 + c^2d^2 + e^2f^2)/(ab^3 + cd^3 + ef^3))^(3/2)`
= `((b^2k^2·b^2 + d^2k^2·d^2 + f^2k^2·f^2)/(bk.b^3 + dk·d^3 + fk·f^3))^(3/2)`
= `[(k^2 (b^4 + d^4 + f^4))/(k (b^4 + d^4 + f^4))]^(3/2)`
= `k^(3/2)`
R.H.S. = `sqrt((ace)/(bdf)) = sqrt((bk·dk·fk)/(bdf)) = k^(3/2)`
L.H.S. = R.H.S.
Hence proved.