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If (P - X) : (Q - X) Be the Duplicate Ratio of P : Q Then Show that `1/P + 1/Q = 1/X` - Mathematics

if (p - x) : (q - x) be the duplicate ratio of p : q then show that `1/p + 1/q = 1/x`

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Solution

We have

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

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

`=> pq^2 - q^2x = p^2q - p^2x`

`=> p^2x - q^2x = p^2q - pq^2`

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

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