A hyperbola H has its centre at the origin and its transverse axis along the x -axis. Two circles, C₁ and C₂ , are drawn with their centres at the right focus of H . The circle C₁ passes through the nearer vertex of H , while C₂ passes through the farther vertex. If the length of the latus rectum of H is 18 units and the ratio of the area of C₂ to the area of C₁ is 9:1 , then the area (in sq. units) of the circle who

Step-by-step solution

Let the equation of the hyperbola be x^2 a^2 - y^2 b^2 = 1 with eccentricity e . The right focus is at (ae, 0) and the vertices are at ( a, 0) . The radius of C₁ (passing through the nearer vertex) is r₁ = ae - a = a(e-1) . The radius of C₂ (passing through the farther vertex) is r₂ = ae - (-a) = a(e+1) . Given the ratio of areas is 9:1 , we have: a^2(e+1)^2 a^2(e-1)^2 = 9 e+1 e-1 = 3 e+1 = 3e-3 2e = 4 e = 2 The length of the latus rectum is given as 18 : 2b^2 a = 18 b^2 = 9a Using the standard relation b^2 = a^2(e