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Question
If `x = (2ab)/(a + b)`, find the value of `(x + a)/(x - a) + (x +b)/(x - b)`.
Solution
`x = (2ab)/(a + b)`
`x/a = (2b)/(a + b)`
Applying componendo and dividendo,
`(x + a)/(x - a) = (2b + a + b)/(2b - a - b)`
`(x + a)/(x - a) = (3b + a)/(b - a)` ...(1)
Also, `x = (2ab)/(a + b)`
Applying componendo and dividendo,
`(x + b)/(x - b) = (2a + a + b)/(2a - a - b)`
`(x + b)/(x - b) = (3a + a)/(a - b)` ...(2)
From (1) and (2)
`(x + a)/(x - a) + (x + b)/(x - b) = (3b + a)/(b - a) + (3a + b)/(a - b)`
`(x + a)/(x - a) + (x + b)/(x - b) = (-3b - a + 3a + b)/(a - b)`
`(x + a)/(x - a) + (x + b)/(x - b) = (2a - 2b)/(a - b) = 2`
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