Question

If x, y, z are in continued proportion, prove that: `(px^2 + qxy + ry^2)/(py^2 + qyz + rz^2) = x/z`.

Answer 1

`x/y = y/z` = k

y = zk

x = yk

= (zk)k

= zk²

L.H.S.

= `(px^2 + qxy + ry^2)/(py^2 + qyz + rz^2)`

= `(p(zk^2)^2 + q(zk^2)(zk) + r(zk)^2)/(p(zk)^2 + q(zk)(z) + rz^2)`

= `(pz^2k^4 + qz^2k^3 + rz^2k^2)/(pz^2k^2 + qz^2k + rz^2)`

= `(z^2k^2(pk^2 + qk + r))/(z^2(pk^2 + qk + r))`

= k²

R.H.S.

= `x/z`

= `(zk^2)/z`

= k²

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