Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci: (0,-4), (0,4); Vertices: (0, -2), (0,2)

A hyperbola is a type of conic section formed by the intersection of a plane and a double cone. It consists of two separate curves called branches, which are mirror images of each other. The standard form of a hyperbola's equation depends on its orientation, which can be horizontal or vertical, determined by the positions of its foci and vertices.

In a hyperbola, the foci are two fixed points located along the transverse axis, which is the line segment that connects the vertices. The vertices are the points where the hyperbola intersects its transverse axis. The distance between the center and each vertex is denoted as 'a', while the distance from the center to each focus is 'c'. The relationship between 'a', 'b' (the distance to the co-vertices), and 'c' is given by the equation c² = a² + b².

The standard form of a hyperbola's equation is expressed as (y²/a²) - (x²/b²) = 1 for a vertical hyperbola, and (x²/a²) - (y²/b²) = 1 for a horizontal hyperbola. In this case, since the foci and vertices are aligned vertically, the equation will take the vertical form. The values of 'a' and 'c' can be derived from the given vertices and foci, allowing for the complete equation to be formulated.