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`
Concept: Concept of Ratio
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