Equation of a hyperbola having foci at (0, 10 ) and passing through the point (2,3) is

Step-by-step solution

The foci of the hyperbola are at (0, 10 ) , which implies the hyperbola is vertical with its center at (0,0) . The equation is of the form y^2 a^2 - x^2 b^2 = 1 . The distance of the foci from the center is ae = 10 , so a^2 e^2 = 10 . For a hyperbola, b^2 = a^2(e^2 - 1) = a^2 e^2 - a^2 . Substituting the known values, b^2 = 10 - a^2 . The hyperbola passes through (2,3) . Substituting these coordinates into the equation: 3^2 a^2 - 2^2 b^2 = 1 , which simplifies to 9 a^2 - 4 10 - a^2 = 1 . Let u = a^2 . Then 9 u - 4