Question

Given : x = `(sqrt(a^2 + b^2)+sqrt(a^2 - b^2))/(sqrt(a^2 + b^2)-sqrt(a^2 - b^2))`
Use componendo and dividendo to prove that `b^2 = (2a^2x)/(x^2 + 1)`.

Answer 1

x = `(sqrt(a^2 + b^2)+sqrt(a^2 - b^2))/(sqrt(a^2 + b^2)-sqrt(a^2 - b^2))`
By componendo and dividendo,
`(x + 1)/(x - 1) = (2sqrt(a^2 + b^2))/(2sqrt(a^2 - b^2)`
Squaring both sides,
`(x^2 + 2x + 1)/(x^2 - 2x + 1) = (a^2 + b^2)/(a^2 - b^2)`
By componendo and dividendo.
`(2(x^2 + 1))/(4x) = (2a^2)/(2b^2)`
⇒ `(x^2 + 1)/(2x) = (a^2)/(b^2)`
⇒ b² = `(2a^2x)/(x^2 + 1)`.
Hence proved.