Let H be a hyperbola with its centre at the origin and transverse axis along the x -axis. Let C₁ be a circle with its centre at the origin that passes through both the foci of H , and let C₂ be the auxiliary circle of H . The area of the region strictly between C₁ and C₂ is 36 sq. units. If the tangents to H at its two vertices intersect the circle C₁ at four distinct points, and the area of the rectangle formed by t

Step-by-step solution

Let the hyperbola be x^2 a^2 - y^2 b^2 = 1 . The circle C₁ passes through the foci ( ae, 0) , so its radius is r₁ = ae . The auxiliary circle C₂ has radius r₂ = a . The area of the region between C₁ and C₂ is: (r₁^2 - r₂^2) = (a^2e^2 - a^2) Since a^2e^2 - a^2 = b^2 , the area is b^2 . Given b^2 = 36 b^2 = 36 b = 6 . The tangents to H at its vertices are the lines x = a and x = -a . The equation of C₁ is x^2 + y^2 = a^2e^2 . Substituting x = a into the equation of C₁ : a^2 + y^2 = a^2e^2 y^2 = a^2(e^2-1) = b^2 y = b