Question

If (p – x) : (q – x) be the duplicate ratio of p : q then show that : `1/p + 1/q = 1/x`.

Answer 1

We have,

`(p -x)/(q - x) = p^2/q^2`

`=>` q²(p – x) = p²(q – x)

`=>` pq² – q²x = p²q – p²x

`=>` p²x – q²x = p²q – pq²

`=>` x(p² – q²) = pq(p – q)

`=>` x(p – q)(p + q) = pq(p – q)

`=> x = (pq)/(p + q)`

`=> (p + q)/(pq) = 1/x`

`=> p/(pq) + q/(pq) = 1/x`

`=> 1/p + 1/q = 1/x`