A Problem on Conics!

Geometry Level 5

What is the exhaustive set of values of \(\displaystyle \alpha^2\) such that there exists a tangent to the ellipse \(\displaystyle x^2+\alpha^2y^2=1 \) and the portion of tangent intercepted by hyperbola \(\displaystyle x^2-\alpha^2y^2=1\) subtends a right angle at the centre of the curves?

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