Question

If a, b, c and d are in proportion, prove that: (a⁴ + c⁴) : (b⁴ + d⁴) = a² c² : b² d².

Answer 1

∵ a, b, c, d are in proportion
`a/b = c/d` = k(say)
a = bk, c = dk.
(a⁴ + c⁴) : (b⁴ + d⁴) = a² c² : b² d² 

L.H.S. = `(a^4 + c^4)/(b^4 + d^4)`

= `(b^4k^4 + d^4k^4)/(b^4 + d^4)`

= `(k^4(b^4 + d^4))/((b^4 + d^4)`
= k⁴
R.H.S. = `(a^2c^2)/(b^2d^2)`

= `(k^2b^2.k^2d^2)/(b^2.d^2)`
= k⁴
Hence L.H.S. = R.H.S.