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Question
If x = `(8ab)/"a + b"` find the value of `(x + 4a)/(x - 4a) + (x + 4b)/(x - 4b)`
Solution
x = `(8ab)/"a + b"`
⇒ `x/(4a) = (2b)/"a + b"`
Applying componendo and dividendo,
`(x + 4a)/(x - 4a) = (2b + a + b)/(2b - a - b) = (3b + a)/(b - a)` ...(i)
Again `x/(4b) = (2a)/"a + b"`
Applying componendo and dividendo,
`(x+ 4b)/(x - 4b) = (2a + a + b)/(2a -a - b) = (3a + b)/(a - b)` ...(ii)
Adding (i) and (ii)
`(x + 4a)/(x - 4a) + (x + 4b)/(x - 4b)`
= `(3b + a)/(b - a) + (3a + b)/(a - b)`
= `-(a + 3b)/(a - b) + (3a + b)/(a - b)`
= `(-a - 3b + 3a + b)/(a - b)`
= `(2a - 2b)/(a - b)`
= `(2(a - b))/(a - b)`
= 2.
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