In a hyperbola, the horizontal distance from the center to the vertices is a=8, the vertical distance from the center to the vertices is b=12, the hyperbola opens up and down, and the center is (0,0). Find the equation of the hyperbola.

The equation of the parabola is given as follows;

Step-by-step explanation:

Here we have the general equation of a vertical hyperbola, where the y term is positive is given by the relation;

Where:

(h, k) are the coordinates of the center which is given as (0, 0)

a = Horizontal distance from the center of the hyperbola = 8

b = Vertical distance from the center of the hyperbola = 12

Plugging in the values, we have the equation of the parabola given as follows;

Hence the equation of the parabola is given as follows;

One of the many points that can possibly be added to the given points to form a cone is (2,2). The additional point will create a cone because the base will pass through (0,0) and (0,4) in form of a circular shape and its only vertex is at (2,2). 

A cone is not formed by the given set points because there are only 2 points and at least 3 points will be needed to form the base of the pyramid.