Let  be the largest circle centered at $\left(2,0\right)$ and inscribed in the ellipse  $\frac{{\mathrm{x}}^2}{36}+\frac{{\mathrm{y}}^2}{16}=1,\text{ If }(1,\mathrm{\alpha })$lies on 5 α² , then  is equal to ________

Equation of normal to the ellipse $\frac{{\mathrm{x}}^2}{36}+\frac{{\mathrm{y}}^2}{16}=1$ at any point
$\mathrm{P}(6\cos \mathrm{\theta },4\sin \mathrm{\theta })\text{ is }3\sec \mathrm{\theta x}-2\mathrm{cosec}\mathrm{\theta }.\mathrm{y}=10$this normal is also the normal of the circle passing through the point 
$\begin{array}{r}(2,0)\Rightarrow 6\sec \mathrm{\theta }=10\text{ or }\sin \mathrm{\theta }=0\left(\text{ not possible }\right)\\ \cos \mathrm{\theta }=\frac{3}{5}\text{ and }\sin \mathrm{\theta }=\frac{4}{5}\therefore \mathrm{P}\left(\frac{18}{5},\frac{16}{5}\right)\end{array}$
The largest radius of the circle  $\mathrm{r}=\frac{\sqrt{320}}{5}$
equation of the circle $(\mathrm{x}-2{)}^2+{\mathrm{y}}^2=\frac{64}{5}$ passing through (1, α)
then ${\mathrm{\alpha }}^2=\frac{59}{5}$