The distance between the foci of a hyperbola is 16 and its eccentricity is 2 . Then its equation is

Step-by-step solution

The distance between the foci of a hyperbola is given by 2ae = 16 . Given the eccentricity e = 2 , we have 2a 2 = 16 , which simplifies to a 2 = 8 , so a = 8 2 = 4 2 . Squaring a , we get a^2 = (4 2 )^2 = 16 2 = 32 . For a hyperbola, the relationship between a, b, and e is b^2 = a^2(e^2 - 1) . Substituting the known values, b^2 = 32(( 2 )^2 - 1) = 32(2 - 1) = 32 . The standard equation of a hyperbola centered at the origin is x^2 a^2 - y^2 b^2 = 1 . Substituting a^2 = 32 and b^2 = 32 , we get x^2 32 - y^2 32 = 1 ,