Given x,y∈R, x2+y2>0. Then the range of x2+y2x2+xy+4y2 is

λ=x2+y2x2+xy+4y2=1+yx21+yx+4yx2=1+z21+z+4z2              (Putting y/x=z)or   λ+λz+4λz2=1+z2z2(4λ−1)+z(λ)+λ−1=0Since z is real,    D≥0∴    λ2−44λ2−5λ+1≥0∴    15λ2−20λ+4≤0∴    λ∈10−4515,10+4515