Let a line L be tangent to the hyperbola x^2 a^2 - y^2 b^2 = 1 . The perpendicular from the origin to L meets it at a point Q . It is given that the locus of Q passes through the point (2 3 , 2) and the eccentricity of the hyperbola is 2 . The distance between the foci of the hyperbola is ______.

Step-by-step solution

The equation of the tangent L to the hyperbola is given by y = mx a^2m^2 - b^2 . The line perpendicular to L passing through the origin has the equation y = - 1 m x , which gives m = - x y . Substituting this into the equation of the tangent, we get: y = (- x y )x a^2 (- x y )^2 - b^2 y^2 + x^2 = a^2x^2 - b^2y^2 Squaring both sides, the locus of Q is: (x^2 + y^2)^2 = a^2x^2 - b^2y^2 Given the eccentricity e = 2 , we have b^2 = a^2(e^2 - 1) = a^2(2 - 1) = a^2 . So the locus equation simplifies to: (x^2 + y^2)^2 = a^