Question

Which are the foci for the hyperbola modeled by the equation (y - 4)²/25 - (x - 2)²/11 = 1?

1) (8, 4) and (8, -5)
2) (0, -5) and (0, 5)
3) (2, 11) and (2, -3)
4) (2, 10) and (2, -2)

Answer 1

The foci of the hyperbola modeled by the equation are (2, 10) and (2, -2).

Step-by-step explanation:

The equation of the hyperbola is given by (y - 4)²/25 - (x - 2)²/11 = 1.

To find the foci of the hyperbola, we need to determine the values of a and b. The equation of a hyperbola in standard form is (y - k)²/a² - (x - h)²/b² = 1, where the center of the hyperbola is at (h, k).

Comparing the given equation with the standard form, we have h = 2, k = 4, a² = 25, and b² = 11.

Therefore, a = 5 and b = √11.

The foci of the hyperbola can be found using the formula c = √(a² + b²), where c is the distance from the center to each focus.

Substituting the values of a and b into the formula, we get c = √(25 + 11) = √36 = 6.

Since the center is at (h, k) = (2, 4), the foci are located at (h, k + c) = (2, 4 + 6) = (2, 10) and (h, k - c) = (2, 4 - 6) = (2, -2).

Therefore, the correct answer is (4) ((2, 10) and (2, -2)).