Question

the foci for the hyperbola (y-2)^2/16 - (x 1)^2/144 = 1 are (-1, 2 4 sqrt 10) and (-1, 2 -4 sqrt 10) a) true b) false

Answer 1

The foci of the hyperbola
`(y-2)^2/16 - (x-1)^2/144 = 1 are (-1, 2+4√(10)) and (-1, 2-4√(10))).`

Step-by-step explanation:

The given equation represents a hyperbola. To find the foci of the hyperbola, we need to determine the distance between the center and the foci. The distance between the center of the hyperbola and the foci, denoted as c, can be found using the formula c = √(a² + b²), where a and b are the lengths of the semi-major and semi-minor axes respectively. In this case, the center of the hyperbola is (-1, 2) and the length of the semi-major axis is 12, so a = 12. The length of the semi-minor axis is 4, so b = 4.

Substituting the values, we have c = √(12² + 4²) = √(144 + 16) = √160 = 4√10.

Therefore, the foci of the hyperbola are (-1, 2 + 4√10) and (-1, 2 - 4√10).

Answer 2

The answer is B. False.

Identify the type of hyperbola: The given equation is in the form
`(y-k)^2/a^2 - (x-h)^2/b^2 = 1`, which represents a hyperbola with a vertical axis. This means the foci will lie on a vertical line.

Find the center: The center of the hyperbola is given by the coordinates (h, k). In this case, h = -1 and k = 2, so the center is (-1, 2).

Find the values of a and b: From the equation, we can see that a^2 = 16 and b^2 = 144. Therefore, a = 4 and b = 12.

Calculate the focal length: The focal length, denoted by c, is related to a and b by the equation
`c^2 = a^2 + b^2`. Substituting the values, we get
`c^2 = 16 + 144 = 160`, so c = 4√10.

Determine the foci: Since the hyperbola has a vertical axis, the foci will be located at a distance of c units above and below the center. Therefore, the foci are at:

(-1, 2 + 4√10)

(-1, 2 - 4√10)

Incorrect statement: The statement in the prompt incorrectly places the foci at the same y-coordinate, (-1, 2 + 4√10) and (-1, 2 - 4√10). This would be true for a hyperbola with a horizontal axis, but not for the given hyperbola with a vertical axis.