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)`.