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Question
If (a – x) : (b – x) be the duplicate ratio of a: b show that:
`(1)/x = (1)/a + (1)/b.`
Solution
Here (a - x) : (b - x) is duplicate ratio of a : b
∴ `a^2/b^2 = (a - x)/(b - x)`
⇒ `a^2b - a^2x = b^2a - b^2x`
⇒ `a^2b - b^2a = a^2x - b^2x`
⇒ ab (a - b) = x (a - b) (a + b)
⇒ ab = x (a + b)
⇒ `(1)/x = (a + b)/(ab)`
⇒ `(1)/x = (1)/a + (1)/b`
Hence proved.
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