Question

Find the equation of the hyperbola with the following properties. Express your answer in standard form.

Foci at (-8, -2) and (-8,-8)
Vertices at (-8, -4) and (-8,-6)

a. ( frac((y+5)^2)(2^2) - frac((x+8)^2)(3^2) = 1 )
b. ( frac((y+8)^2)(2^2) - frac((x+8)^2)(3^2) = 1 )
c. ( frac((y+5)^2)(3^2) - frac((x+8)^2)(2^2) = 1 )
d. ( frac((y+8)^2)(3^2) - frac((x+8)^2)(2^2) = 1 )

Answer 1

The equation of the hyperbola is (y + 5)^2/2^2 - (x + 8)^2/3^2 = 1.

Step-by-step explanation:

The equation of a hyperbola can be expressed in the standard form:

(y - k)2/a2 - (x - h)2/b2 = 1

where (h, k) represents the center of the hyperbola.

Given that the foci are (-8, -2) and (-8, -8) and the vertices are (-8, -4) and (-8, -6), we can determine that the center of the hyperbola is at (-8, -5).

Since the hyperbola is centered at (-8, -5) and the vertices are 2 units from the center vertically, we have a value of a = 2. Similarly, the foci are 3 units from the center vertically, giving us a value of c = 3.

Plugging these values into the standard form, we get the equation:

(y + 5)2/22 - (x + 8)2/32 = 1