Let  ${\text{5x}}^{\text{2}}{\text{+4xy+y}}^{\text{2}}{\text{=1\hspace{0.17em}\&\hspace{0.17em}z}}^{\text{2}}{\text{=x}}^{\text{2}}{\text{+y}}^{\text{2}}\text{,x>y>0}$. Which of the following is/are INCORRECT?

Let   $x=r\cos \theta \&y=r\sin \theta ,0<\theta <\frac{\pi }{4}$ , then z=r and
 $5r^2{\cos }^2\theta +4r^2\sin \theta \cos \theta +r^2{\sin }^2\theta \Rightarrow r^2=\frac{1}{2\cos 2\theta +2\sin 2\theta +3}$
 $\Rightarrow r\frac{dr}{d\theta }=2\frac{\sin 2\theta -\cos 2\theta }{{(2\cos 2\theta +2\sin 2\theta +3)}^2}$
Now r is increasing if  $\sin 2\theta >\cos 2\theta \Rightarrow \tan 2\theta >1\Rightarrow \frac{\pi }{8}<\theta <\frac{\pi }{4}$.
And r is decreasing if  $\sin 2\theta <\cos 2\theta \Rightarrow \tan 2\theta <1\Rightarrow 0<\theta <\frac{\pi }{8}$.
Now for  $\theta =\frac{\pi }{8},r^2=\frac{1}{3+2\sqrt{2}}$  i.e. $(3-2\sqrt{2})$  hence  $r=\sqrt{2}-1$.