The equation of an ellipse is given by $5{\mathrm{x}}^2+5{\mathrm{y}}^2-6\mathrm{xy}-8=0$. If $r_1, r_2$are distances of points on the ellipse which are at maximum & minimum distance from origin then ${\mathrm{r}}_1+{\mathrm{r}}_2=$

The equation of an ellipse is given by $5{\mathrm{x}}^2+5{\mathrm{y}}^2-6\mathrm{xy}-8=0$
Any point on ellipse at a distance r from origin is $(\mathrm{rcos\theta },\mathrm{rsin\theta })$
$$\begin{gathered} \Rightarrow 5r^2-6r^2\sin \theta \cos \theta =8 \\ \Rightarrow r^2(5-3(2\sin \theta \cos \theta \left)\right)=8 \end{gathered}$$
$\Rightarrow {\mathrm{r}}^2=\frac{8}{5-3\sin 2\mathrm{\theta }}$
 is maximum if $5-3\sin 2\mathrm{\theta }$ is minimum $\Rightarrow {\mathrm{r}}^2=4$
${\mathrm{r}}^2\mathrm{m}$ in if $(5-3\sin 2\mathrm{\theta })$ is maximum$=8\Rightarrow {\mathrm{r}}^2=1$
${\mathrm{r}}_1+{\mathrm{r}}_2=2+1=3$