(*) 1+N=.....=x²/(x-2)²+4

We have x² \(\ge\) 0 , (x-2)²+4 \(\ge\) 4 > 0 

So 1+N \(\ge\) 0 => N \(\ge\) -1 ;equality : x=0 

(*)1-N=....=(x-4)²/(x-2)²+4 

....... -> N \(\le\) 1 , equality : x=4

Using cauchy-schwarz inequality , we have:

A(1²+1²)=(4x²+y²)(1²+1²)=[(2x)²+y²].(1²+1²) \(\ge\) (2x.1+y.1)² = 6²=36

=> A \(\ge\) 36/2=18

minA=18, equality : x=1,5;y=3