Question

Find x from the following equations : `(sqrt(1 + x) + sqrt(1 - x))/(sqrt(1 + x) - sqrt(1 - x)) = a/b`

Answer 1

`(sqrt(1 + x) + sqrt(1 - x))/(sqrt(1 + x) - sqrt(1 - x)) = a/b`
Applying componendo and dividendo,

`(sqrt(1 + x) + sqrt(1 - x) + sqrt(1 + x) - sqrt(1 - x))/(sqrt(1 + x) + sqrt(1 - x) - sqrt(1 + x) + sqrt(1 - x)) = (a + b)/(a - b)`

⇒ `(2sqrt(1 + x))/(2sqrt(1 - x)) = (a + b)/(a - b)`

⇒ `(sqrt(1 + x))/(sqrt(1 - x)) = (a + b)/(a - b)`

Squaring both sides,
`(1 + x)/(1 - x) = (a + b)^2/(a - b)^2`

Again applying componendo and dividendo,
`(1 + x + 1 - x)/(1 + x - 1 + x)`

= `((a + b)^2 + (a - b)^2)/((a + b)^2 - (a - b)^2)`

⇒ `(2)/(2x) = (2(a^2 + b^2))/(4ab)`

⇒ `(1)/x = (a^2 + b^2)/(2ab)`

∴ x = `(2ab)/(a^2 + b^2)`.