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Question

The set of values of α2, if there exists a tangent to the ellipse x2α2+y2=1 such that the portion of the tangent intercepted by the hyperbola α2x2y2=1 subtends a right angle at the centre of the curves, is

A
[5+12,2]
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B
[1,2]
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C
[512,1]
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D
[512,5+12]
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Solution

The correct option is A [5+12,2]
Equation of any tangent to x2α2+y2=1 in slope form is, y=mx±α2m2+1
(ymxα2m2+1)2=1
By homogenisation, we can write
(α2x2y2)(α2m2+1)=y2+m2x22mxy
The portion of tangent subtends 90 at the origin.
coefficient of x2+coefficient of y2=0
(α2m2+1)(α21)=1+m2
α2(α21)m2+α21=1+m2
[α2(α21)1]m2=2α2
m2=2α2α4α210
α22α4α210

Let α2=t0
Then t2t2t10
t[1+52,2]

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