Question

Find x from the equation: `(a+ x + sqrt(a^2 - x^2))/(a + x - sqrt(a^2 - x^2)) = b/x`

Answer 1

`(a+ x + sqrt(a^2  x^2))/(a + x - sqrt(a^2 - x^2)) = b/x`
Applying componendo and dividendo,

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

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

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

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

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

⇒ `(a + x)/(a - x) = (b + x)^2/(b - x)^2`
Again applying componendo and dividendo,

`(a + x + a - x)/(a + x - a + x)`

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

⇒ `(2a)/(2x) = (2(b^2 + x^2))/(4bx)`

⇒ `a/x = (b^2 + x^2)/(2bx)`
2abx = x(b² + x²)
⇒  2ab = b² + x²
⇒  x² = 2ab – b²
x = `sqrt(2ab - b^2)`

Answer 2

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

Simplify the Left-Hand Side,

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

Factoring `sqrt(a + x)` from the numerator and denominator:

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

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

Apply componendo and dividendo,

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

Square both sides,

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

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

Apply componendo and dividendo again,

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

`(2a)/(2x) = (2(b^2 + x^2))/(4bx)`

`a/x = (b^2 + x^2)/(2bx)`

2abx = x(b² + x²)

2ab = b² + x²

x² = 2ab − b²

x = ±`sqrt(2ab - b^2)`