The exhaustive set of values of ${\alpha }^2$ such that there exists a tangent to the ellipse $x^2+{\alpha }^2{y}^2={\alpha }^2$ and the portion of tangent intercepted by hyperbola ${\alpha }^2{x}^2-y^2=1$ subtends a right angle at the centre of the curves is

$x^2+{\alpha }^2{y}^2={\alpha }^2$

Equation of tangent is $\frac{x\cos \theta }{\alpha }+\sin \theta y=1$
Tangent intersects the hyperbola at A and B such that $ACB=\frac{\pi }{2}$  
Where C is centre of hyperbola

 ${\alpha }^2{x}^2-y^2=1{\left(1\right)}^2$
 ${\alpha }^2{x}^2-y^2={(\frac{x\cos \theta }{\alpha }+\sin \theta y)}^2$

$\Rightarrow x^2({\alpha }^2-\frac{{\cos }^2\theta }{{\alpha }^2})+y^2(-1-{\sin }^2\theta )-2xy\frac{\cos \theta \sin \theta }{\alpha }=0$ ___ (1)
(1) represents of equation of pair of lines OA and OB
Coefficient of $x^2+$ coefficient of $y^2=0$
$\Rightarrow -1-\frac{{\cos }^2\theta }{{\alpha }^2}+{\alpha }^2-{\sin }^2\theta =0$ 
$\Rightarrow {\alpha }^2=\frac{1+{\sin }^2\theta \pm \sqrt{5+{\sin }^4\theta -2{\sin }^2\theta }}{2}={\alpha }^2\in [\frac{\sqrt{5}+1}{2},2]$