An Angle at the Focus

Difficulty: Hard · Topics: Geometry, Conic Sections, Hyperbola

Fokusdagi burchak

The circle having the two foci F and F' of the hyperbola dfracx^24-y^2=1 as the endpoints of a diameter meets the hyperbola in the first quadrant at the point P. Writing angle PFF'=θ, we have tanθ=m+n√6.

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Find the value of m^2+n^2, where m and n are rational numbers. (Here F is the focus with positive x-coordinate.)