Comprehension Consider a hyperbola \( x y=4 \) and a line \( y+2 x=4, O \) is the centre of hype...

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Comprehension
Consider a hyperbola \( x y=4 \) and a line \( y+2 x=4, O \) is the centre of hyperbola. Tangent at any point \( P \) of hyperbola intersect the coordinate axes at \( A \) and \( B \).

Shortest distance between the line and hyperbola is
(a) \( \frac{8 \sqrt{2}}{\sqrt{5}} \)
(b) \( \frac{4(\sqrt{2}-1)}{\sqrt{5}} \)
(c) \( \frac{2 \sqrt{2}}{\sqrt{5}} \)
(d) \( \frac{4(\sqrt{2}+1)}{\sqrt{5}} \)
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