Write the equation of the hyperbola that has a center at (4, - 1), a focus at (11, - 1), and a vertex at (0, - 1).

Answer: 56

Step-by-step explanation:

Final answer:

Given the center, focus, and vertex of a hyperbola, the equation of the hyperbola can be determined using the standard formula for a hyperbola and calculations for the values of a and b. For the hyperbola with center (4, -1), focus (11, -1), and vertex (0, -1), the equation is (x - 4)²/16 - (y + 1)²/33 = 1.

Explanation:

The subject of the question is to write the equation of the hyperbola given the center, focus, and vertex. In general, the equation of a horizontal hyperbola is (x - h)²/a² - (y - k)²/b² = 1 where the (h, k) is the center, a is the distance from the center to a vertex, and b is the distance from the center to a co-vertex. In this case, the center is (4, -1), the focus is (11, -1), the and vertex is (0, -1).

To determine a, calculate the distance from the center to a vertex. With the center at (4, -1) and vertex at (0, -1), a = 4. To determine b, apply the hyperbola's relationship of c² = a² + b², where c is the distance from the center to a focus. Given that the distance to the focus (from (4, -1) to (11, -1)) is 7 (so, c = 7) and a = 4, solve for b to get b = sqrt(c² - a²) = sqrt(49-16)= sqrt(33). Therefore, the equation of the hyperbola is (x - 4)²/16 - (y + 1)²/33 = 1.

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