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^3 + 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) = (sqrt(a^2 + b^2) + sqrt(a^2 - b^2) + sqrt(a^2 + b^2) - sqrt(a^2 - b^2))/(sqrt(a^2 + b^2) + sqrt(a^2 - b^2) - sqrt(a^2 + b^2) + sqrt(a^2 - b^2))`

(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 dividend

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

`=> (2(x^2 + 1))/"4x" = (2a^2)/(2b^2)`

`=> (x^2 + 1)/(2x) = a^2/b^2`

`=> b^2 = (2a^2x)/(x^2 + 1)`

Hence Proved