A Problem on Conics!

Geometry Level 5

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

[ 5 − 1 2 , 1 ) \displaystyle \left [\frac{\sqrt{5}-1}{2},1 \right ) ( 1 , 2 ] \displaystyle (1,2] [ 5 − 1 2 , 1 ) ∪ ( 1 , 5 + 1 2 ] \displaystyle \left[\frac{\sqrt{5}-1}{2},1 \right ) \cup \left (1,\frac{\sqrt{5}+1}{2} \right] [ 5 + 1 2 , 2 ] \displaystyle \left [\frac{\sqrt{5}+1}{2},2 \right ]

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Saarthak Marathe by Saarthak Marathe , 22, India

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