# If  2x = sqrt a +1/sqrt a then what is the value of sqrt(x^2 - 1)/{x - sqrt(x^2 - 1)} ?

Jan 7, 2015

I tried by writing an expression for x only:
x=1/2×(sqrt(a)+1/(sqrt(a)))
I then substituted it in the expression:
$\frac{\sqrt{{x}^{2} - 1}}{x - \sqrt{{x}^{2} - 1}}$
Considering that:
sqrt(x^2-1)=sqrt(1/4×(sqrt(a)+1/(sqrt(a)))^2-1)=
=sqrt((a^2-2a+1)/(4a)=
$= \frac{a - 1}{2 \sqrt{a}}$

So that $\frac{\sqrt{{x}^{2} - 1}}{x - \sqrt{{x}^{2} - 1}}$ becomes:

$\frac{a - 1}{2 \sqrt{a}} / \left(\frac{\sqrt{a}}{2} + \frac{1}{2 \sqrt{a}} - \frac{a - 1}{2 \sqrt{a}}\right) =$
$= \frac{a - 1}{2}$