Question

If x = `(2a + b)/(a + b)` find the value of `(x + a)/(x - a) + (x + b)/(x - b)`

Answer 1

x = `(2a + b)/"a + b"`

⇒ `x/a = (2b)/"a + b"`

Applying componendo and dividendo,

`"x + a"/"x - a" = (2b + a + b)/(2b - a - b) = (3b + a)/"b - a"`     ...(i)

Again `x/b = (2a)/"a + b"`
Applying componendo and dividendo,

`"x + b"/"x - b" = (2a + a + b)/(2a - a - b) = (3a + b)/"a - b"`   ...(ii)

Adding (i) and (ii)
`"x + a"/"x - a" + "x + b"/"x - b"`

= `(3b + a)/"b - a" + (3a + b)/"a - b"`

= `-(a + 3b)/"a - b" + (3a + b)/"a - b"`

= `(-a - 3b + 3a + b)/"a - b"`

= `(2a - 2b)/"a - b"`

= `(2(a - b))/"a - b"`
= 2.