Question

Find x from the following equations : `(sqrt(a + x) + sqrt(a - x))/(sqrt(a + x) - sqrt(a - x)) = c/d`

Answer 1

`(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)`.