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

1 Answer
Gió
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:
#(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)=#
#=(a-1)/(2sqrt(a))#

So that #(sqrt(x^2-1))/(x-sqrt(x^2-1))# becomes:

#(a-1)/(2sqrt(a))/(sqrt(a)/2+1/(2sqrt(a))-(a-1)/(2sqrt(a)))=#
#=(a-1)/2#

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