$\frac{{\mathrm{x}}^2}{{\mathrm{r}}^2-\mathrm{r}-6}+\frac{{\mathrm{y}}^2}{{\mathrm{r}}^2-6\mathrm{r}+5}=1$ will represents the ellipse, if r lies in the interval

$\begin{array}{l}{\mathrm{r}}^2-\mathrm{r}-6>0\text{ and }{\mathrm{r}}^2-6\mathrm{r}+5>0\\ \Rightarrow (\mathrm{r}-3)(\mathrm{r}+2)>0\text{ and }(\mathrm{r}-1)(\mathrm{r}-5)>0\\ \Rightarrow (\mathrm{r}<-2\text{ or }\mathrm{r}>3)\text{ and }(\mathrm{r}<1\text{ or }\mathrm{r}>5)\\ \Rightarrow \mathrm{r}<-2\text{ or }\mathrm{r}>5\\ \text{ Also }{\mathrm{r}}^2-\mathrm{r}-6\ne {\mathrm{r}}^2-6\mathrm{r}+5\Rightarrow \mathrm{r}\ne \frac{11}{5}.\end{array}$