If maximum distance of any point on the ellipse ${\mathrm{x}}^2+2{\mathrm{y}}^2+2\mathrm{xy}=1$ from its centre be r, then r is equal to 

Here centre of the ellipse is (0, 0) 
Let $\mathrm{P}(\mathrm{r} \cos \mathrm{\theta },\mathrm{r} \sin \mathrm{\theta })$ be any point on the given ellipse 

 $$\begin{gathered} {\mathrm{r}}^2{\cos }^2 \mathrm{\theta }+2{\mathrm{r}}^2{\sin }^2 \mathrm{\theta }+2{\mathrm{r}}^2\mathrm{sin\theta } \cos \mathrm{\theta }=1 \\ \Rightarrow r^2=\frac{1}{{\cos }^2\theta +2{\sin }^2\theta +\sin 2\theta } \\ =\frac{1}{1+{\sin }^2\theta +\sin 2\theta } \\ = \frac{2}{1-\cos 2\theta +2+2\sin 2\theta } \\ =\frac{2}{3-\cos 2\theta +2\sin 2\theta } \\ \Rightarrow r_{max}=\frac{\sqrt{2}}{\sqrt{3}-\sqrt{5}} \end{gathered}$$
$\Rightarrow {\mathrm{r}}_{max}=\frac{\sqrt{2}}{\sqrt{3-\sqrt{5}}}$