Final answer:
The given foci for the hyperbola are incorrect because the correct foci should have an x-coordinate of 1, not -1. The correct foci are (1, 2 ± 4√10). Thus, the statement is False.
Step-by-step explanation:
The question asks whether the given foci (-1, 2 + 4√10) and (-1, 2 - 4√10) are true for the hyperbola with the equation (y-2)²/16 - (x-1)²/144 = 1. To determine the correctness of this statement, we need to understand the general form of the hyperbola equation and how the foci are calculated.
For a hyperbola centered at (h, k) with a vertical transverse axis, the general form is (y-k)²/a² - (x-h)²/b² = 1. The distance between the center and the foci along the vertical transverse axis is denoted c, where c is found using the equation c² = a² + b².
In the given equation, a² = 16 and b² = 144, so c is calculated as c = √(16 + 144) = √160 = 4√10. The center of the hyperbola is at (h, k) = (1, 2). Therefore, the foci should be at (1, 2 ± 4√10), which means that the given foci are incorrect because the x-coordinate should be 1, not -1. Hence, the statement is False.