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

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$[\frac{\sqrt{5} + 1}{2}, 2]$

$(1, 2]$

$[\frac{\sqrt{5} - 1}{2}, 1)$

$[\frac{\sqrt{5} - 1}{2}, 1) \cup (1, \frac{\sqrt{5} + 1}{2}]$

answer is A.

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Detailed Solution

Equation of tangent $P (α \cos θ, \sin θ)$ is
$\frac{x}{α} \cos θ + \frac{y}{1} \sin θ = 1...... (1)$
Homogenizing hyperbola with using
$\Rightarrow α^{2} x^{2} - y^{2} - {(\frac{x}{α} \cos θ + y \sin θ)}^{2} = 0$
Now $a + b = 0 \Rightarrow α^{2} - \frac{\cos^{2} θ}{α^{2}} - 1 - \sin^{2} θ = 0$
$\begin{array}{l} \cos^{2} θ = \frac{α^{2} (2 - α^{2})}{α^{2} - 1} & 0 \leq \cos^{2} θ \leq 1 \\ \Rightarrow α^{2} \in [\frac{\sqrt{5} + 1}{2}, 2] \end{array}$