Question

If b is the mean proportional between a and c, show that `(a^4 + a^2b^2 + b^4)/(b^4 + b^2c^2 + c^4) = a^2/c^2`.

Answer 1

`a/b = b/c`

b² = ac

L.H.S.

= `(a^4 + a^2b^2 + b^4)/(b^4 + b^2c^2 + c^4)`

= `(a^4 + a^2(ac) + (ac)^2)/((ac)^2 + (ac)c^2 + c^4)`

= `(a^4 + a^3c + a^2c^2)/(a^2c^2 + ac^3 + c^4)`

= `(a^2(a^2 + ac + c^2))/(c^2(a^2 + ac + c^2))`

= `a^2/c^2`

= R.H.S.