Explain why it is not possible for a hyperbola to have foci at (0,-2) and (0,2) and vertices at (0,-3) and (0,3).

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 can be expressed as (y^2/a^2) - (x^2/b^2) = 1 for vertical hyperbolas, where 'a' represents the distance from the center to the vertices, and 'c' represents the distance from the center to the foci.

In a hyperbola, the distance from the center to the foci (denoted as 'c') must always be greater than the distance from the center to the vertices (denoted as 'a'). This relationship is expressed mathematically as c^2 = a^2 + b^2. If the foci and vertices are positioned incorrectly, it can lead to contradictions in this fundamental relationship, making the configuration impossible.

For a hyperbola centered at the origin with vertical transverse axis, the foci and vertices must lie along the same line, specifically the y-axis in this case. Given the foci at (0,-2) and (0,2) and vertices at (0,-3) and (0,3), the distance from the center to the foci is 2, while the distance to the vertices is 3. This violates the necessary condition that the distance to the foci must exceed that to the vertices, confirming that such a hyperbola cannot exist.