Question

If a, b, c are in continued proportion, prove that: a : c = (a² + b²) : (b² + c²)

Answer 1

As a, b, c, are in continued proportion
Let `a/b = b/c` = k
a : c = (a² + b²) : (b² + c²)

⇒ `a/c = (a^2 + b^2)/(b^2 + c^2)`

L.H.S. = `a/c`

= `(ck^2)/c`
= k²
R.H.S. = `((ck^2)^2 + (ck)^2)/((ck)^2 + c^2)`

= `(c^2k^4 + c^2k^2)/(c^2k^2 + c^2)`

= `(c^2k^2(k^2 + 1))/(c^2(k^2 + 1)`
=k²
∴ L.H.S. = R.H.S.