Find an equation for the conic section with the given proper

Find an equation for the conic section with the given properties. The hyperbola with foci F1(3, 15) and F2(3, 15) that passes through the point (3, 12)

Solution

as the foci are symmetric about x -axis , so center of hyperbola is (3,0)

also equation is of type (y-k)²/a² - (x-h)²/b² = 1

k = 0, h = 3

a² + b² = 15² = 225

=>    y²/a² - (x-3)²/(225- a²) = 1

  12²/a² - (3-3)²/(225- a²) = 1 => a = 12

b² = 81

=> y²/144 - (x-3)²/81 = 1