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Question
If a: b with a ≠ b is the duplicate ratio of a + c: b + c, show that c2 = ab.
Solution
The duplicate ratio of
`a/b = ((a + c)^2)/((b + c)^2)`
⇒ `a/b = (a^2 + c^2 + 2ac)/(b^2 + c^2 + 2bc)`
⇒ ab2 + ac2 + 2abc = a2b + bc2 + 2abc
⇒ ac2 - bc2 = a2b - ab2
⇒ c2(a - b) = ab(a - b)
Hence, c2 = ab.
Hence proved.
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