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Question

Let A(secθ,2tanθ) and B(secϕ,2tanϕ), where θ+ϕ=π2, be two point on the hyperbola 2x2y2=2. If (α,β) is the point of the intersection of the normals to the hyperbola at A and B, then (2β)2 is equal to

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Solution

Point A(secθ,2tanθ) lies on the hyperbola 2x2y2=2.
2sec2θ4tan2θ=2
2+2tan2θ4tan2θ=2
tan2θ=0
θ=πn, nZ

Similarly, point B(secϕ,2tanϕ) lies on the hyperbola 2x2y2=2.
ϕ=πn, nZ

But in question, it is given that θ+ϕ=π2, which is not possible.
Hence, it is a Bonus question.

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