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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we know,
if a² : b² is the duplicate ratio of a : b
now a/c to question,
(p -x) : (q - x) is the duplicate ratio of p : q
so, from above rule,
(p -x ) : (q - x ) = p² : q²
(p -x)/(q -x ) = p²/q²
q²(p - x) = p²(q - x)
q²p -q²x = p²q -p²x
q²p - p²q = q²x - p²x
pq(q - p) = (q - p)(q + p)x
pq = (q + p)x
pq/x = (q + p)
1/x = (q + p)/pq = q/pq + p/pq
1/x = 1/p + 1/q
hence, 1/x = 1/p + 1/q
hence proved
if a² : b² is the duplicate ratio of a : b
now a/c to question,
(p -x) : (q - x) is the duplicate ratio of p : q
so, from above rule,
(p -x ) : (q - x ) = p² : q²
(p -x)/(q -x ) = p²/q²
q²(p - x) = p²(q - x)
q²p -q²x = p²q -p²x
q²p - p²q = q²x - p²x
pq(q - p) = (q - p)(q + p)x
pq = (q + p)x
pq/x = (q + p)
1/x = (q + p)/pq = q/pq + p/pq
1/x = 1/p + 1/q
hence, 1/x = 1/p + 1/q
hence proved
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