Question

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

Answer 1

Given a, b and c are in continued proportion

=> a : b = b : c

Let `a/b = b/c = k` (say)

=> a = bk, b = ck

Now L.HS = `a/c = ck^2/c = k^2`

R.HS = `(a^2 + b^2)/(b^2 + c^2)`

`= ((ck^2)^2 + (ck)^2)/((ck)^2 + c^2)`

`= (c^2k^2(k^2 + 1))/(c^2(k^2 + 1))`

`= k^2`

LHS = RHS