Question

If x = `(sqrt(a + 1) + sqrt(a - 1))/(sqrt(a + 1 - sqrt(a - 1)`, using properties of proportion, show that x² – 2ax + 1 = 0

Answer 1

We have x = `(sqrt(a + 1) + sqrt(a - 1))/(sqrt(a + 1 - sqrt(a - 1)`

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

(Applying componendo and dividendo)

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

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

(Again applying componendo and dividendo)

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

⇒ `(2x^2 + 2)/(4x)` = a

⇒ `(2(x^2 + 1))/(4x)` = a

⇒ `((x^2 + 1))/(2x)` = a

⇒ 2ax = x² + 1

⇒ x² – 2ax + 1 = 0

Proved.