If a = (√3 + 1)/(√3 – 1) and b = (√3 – 1)/(√3 + 1) then (a² + ab + b²)/(a² – ab + b²) = ?

Explanation

Rationalising,

a=(√3+1)/(√3-1)=2+√3,

b=(√3-1)/(√3+1)=2-√3.

Hence:

a+b=4, ab=1.

So

a^2+b^2=(a+b)^2-2ab=16-2=14.

Thus

(a^2+ab+b^2)/(a^2-ab+b^2)
=
(14+1)/(14-1)
=
15/13.