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Question
Find x from the following equations : `(sqrt(a + x) + sqrt(a - x))/(sqrt(a + x) - sqrt(a - x)) = c/d`
Solution
`(sqrt(a + x) + sqrt(a - x))/(sqrt(a + x) - sqrt(a - x)) = c/d`
Applying componendo and dividendo,
`(sqrt(a + x) + sqrt(a - x) + sqrt(a + x) - sqrt(a - x))/(sqrt(a+ x) + sqrt(a - x) - sqrt(a + x) + sqrt(a - x)) = (c + d)/(c - d)`
⇒ `(2sqrt(a + x))/(2sqrt(a - x)) = (c + d)/(c - d)`
⇒ `sqrt(a + x)/(sqrt(a - x)) = (c + d)/(c - d)`
Squaring both sides
`(a + x)/(a - x) - (c + d)^2/(c - d)^2`
Again applying componendo and dividendo
`(a + x + a - x)/(a + x - a + x) = ((c + d)^2 + (c - d)^2)/((c + d)^2 - (c - d)^2`
⇒ `(2a)/(2x) = (2(c^2 + d^2))/(4cd)`
⇒ `a/x = (c^2 + d^2)/(2cd)`
⇒ x(c2 + d2) = 2acd
⇒ x = `(2acd)/(c^2 + d^2)`.
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