# For complex lovers

**Algebra**Level pending

Let \(\theta=|max(arg(Re(\frac{1}{z-i}).Re(z)+iIm(\frac{1}{z-i})))|\)

where \(|z|=1\) and \(z \neq i\) and \(w=e^{i \theta}\),

if \(x=aw+bw^2+c , y=a+bw+cw^2 ,z=aw^2+b+cw\) then

Find the value of \(|\frac{x^2+y^2+z^2+2ab+2bc+2ac}{a^2+b^2+c^2}|\)

(where \(a,b,c,x,y,z \) are non zero complex numbers).