If maximum distance of any point on the ellipse $x^2+2y^2+2xy=1$ from its Centre $C(0,0)$is $r$is equal to

Here Centre of the ellipse is $(0,0)$

Let $P(r\cos \theta ,r\sin \theta )$be any point on the given ellipse then

$r^2{\cos }^2\theta +2r^2{\sin }^2\theta +2r^2\sin \theta \cos \theta =1$

$\Rightarrow r^2=\frac{1}{{\cos }^2\theta +2{\sin }^2\theta +\sin 2\theta }$

$=\frac{1}{{\sin }^2\theta +1+\sin 2\theta }$

$=\frac{2}{1-\cos 2\theta +2+2\sin 2\theta }$

$=\frac{2}{3-\cos 2\theta +2\sin 2\theta }\Rightarrow r_{\max }=\frac{\sqrt{2}}{\sqrt{3-\sqrt{5}}}$