Final answer:
The foci of the hyperbola modeled by the equation (y-4)^2/25 - (x-2)^2/11 = 1 are located at (2, 10) and (2, -2).
Step-by-step explanation:
The student is asking about the foci of a hyperbola modeled by the equation (y-4)2/25 - (x-2)2/11 = 1. Like an ellipse, a hyperbola has two foci. However, unlike an ellipse, the distances from the foci to any point on the hyperbola have a constant difference, not sum. The foci of a hyperbola can be found using the equation c2 = a2 + b2, where c is the distance from the center of the hyperbola to one of its foci, a is the distance from the center to the vertices along the transverse axis, and b is the distance from the center to the vertices along the conjugate axis.
In this case, since the hyperbola is aligned along the vertical axis, a2 = 25 and b2 = 11, therefore c2 = 25 + 11 = 36. Taking the square root of both sides, we find that c = 6. The center of the hyperbola is (2, 4), so the foci are located at (2, 4 + c) and (2, 4 - c), or (2, 10) and (2, -2).